Proof. This follows from [1] which is stated for an algebraic space \(S\): the case of an arbitrary stack \(S\) follows by taking a smooth atlas. ◻

Proof. The equivalence \(\eqref{T:VcJ-nod1}\Longleftrightarrow \eqref{T:VcJ-nod2}\) is shown in [2].

The implication \(\eqref{T:VcJ-nod3}\Longrightarrow \eqref{T:VcJ-nod2}\) follows from the fact that any \(1\)-parameter smoothing of \(X\) is a pull-back of the semiuniversal deformation of \(X\).

The implication \(\eqref{T:VcJ-nod1}\Longrightarrow \eqref{T:VcJ-nod3}\) follows from [1]. ◻

Proof. Assume first that Condition [Hyp1] is satisfied. Then, any geometric fiber \((\overline{\mathcal{J}}_{X/S}^{\chi})_s\), for \(s\) a geometric point of \(S\), is a universally smoothable compactified Jacobian of \(X_s\) in the sense of Theorem 2[T:VcJ-nod3]. Hence we can apply Theorem 2 in order to get a unique V-stability condition \(\mathfrak{s}^s\) of characteristic \(\chi\) on \(X_s\) such that \[\label{E:fiberJXS} (\overline{\mathcal{J}}_{X/S}^{\chi})_s=\overline{\mathcal{J}}_{X_s}(\mathfrak{s}^s).\tag{8}\]

Assume now that Condition [Hyp2] is satisfied. Because \(S^{\operatorname{sm}}\) is dense in \(S\), for any geometric point \(s\) of \(S\) we can find a morphism \(f: \Delta=\operatorname{Spec}R\to S\), where \(R\) is a DVR, such that the closed point of \(\Delta\) is \(s\) and the generic point of \(\Delta\) is mapped to a point of \(S^{\operatorname{sm}}\). In particular, the pull-back \(X_{\Delta}/\Delta\) of the family \(X/S\) along \(f\) is a one-parameter smoothing of \(X_s\). By pulling back \(\overline{\mathcal{J}}_{X/S}^\chi\) along \(f\), we get a fine compactified Jacobian stack \(\overline{\mathcal{J}}_{X_\Delta/\Delta}^\chi\) for \(X_{\Delta}/\Delta\) whose central fiber is \((\overline{\mathcal{J}}_{X/S}^{\chi})_s\). This implies that \((\overline{\mathcal{J}}_{X/S}^{\chi})_s\) is a fine compactified Jacobian stack that is weakly smoothable according to the terminology of [15]. Hence, we can apply [15] in order to deduce that there exists a unique V-stability condition \(\mathfrak{s}^s\) of characteristic \(\chi\) on \(X_s\) such that 8 holds true also in this case.

By 8 and by [1], we deduce that \(\mathfrak{s}=\{\mathfrak{s}^s\}\) is a V-stability condition on \(X/S\) of characteristic \(\chi\) such that \[\overline{\mathcal{J}}_{X/S}^{\chi}=\overline{\mathcal{J}}_{X/S}(\mathfrak{s}),\] and we are done. ◻

Proof. The fact that \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s})\) is a compactified universal Jacobian stack, as well as the fact that it is fine if and only if \(\mathfrak{s}\) is general, follows from Fact 1. The bijectivity of the map follows from Theorem 3, using that \(\overline{\mathcal{C}}_{g,n}/ \overline{\mathcal{M}}_{g,n}\) is versal at any point. Finally, it remains to observe that \[\mathfrak{s}_1\geq \mathfrak{s}_2\Leftrightarrow \overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\subseteq \overline{\mathcal{J}}_{g,n}(\mathfrak{s}_2).\] Indeed, the implication \(\Rightarrow\) follows from the definition of \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_i)\). The implication \(\Leftarrow\) follows from the above mentioned definition and [1]. ◻

Proof. Let us first show that the map is well-defined, i.e. that \(\mathfrak{s}^{\sigma}\in \operatorname{VStab}^{\chi}_{g,n}\). The function \(\mathfrak{s}^\sigma(X)\) is a V-stability on \(X\) of characteristic \(\chi\), i.e. \(\mathfrak{s}^\sigma(X)\) satisfies the two properties of [1], as it follows from the fact that \(\sigma\) satisfies the two properties [E:condUni1] and [E:condUni2] of Definition 5 and by using that \(\operatorname{type}(Y^c)=\operatorname{type}(Y)^c\) and that, if \(X=Y_1\cup Y_2\cup Y_3\) with \(Y_1, Y_2, Y_3 \in \operatorname{BCon}(X)\) have no pairwise common irreducible components, then \([\operatorname{type}(Y_1, \operatorname{type}(Y_2), \operatorname{type}(Y_3))]\) is a triangle in \(\mathbb{D}_{g,n}\). Moreover, the collection \(\mathfrak{s}^{\sigma}=\{\mathfrak{s}^\sigma(X)\}_{X\in \overline{\mathcal{M}}_{g,n}}\) is a V-stability of type \((g,n)\) since, for any étale specialization \(\xi: X_1\rightsquigarrow X_2\) of geometric points of \(\overline{\mathcal{M}}_{g,n}\) and any \(Y\in \operatorname{BCon}(X_1)\), we have that \(\operatorname{type}(Y)=\operatorname{type}(\xi_*(Y))\), which then implies that \(\xi^*(\mathfrak{s}(X_2))=\mathfrak{s}(X_1)\).

We will now define a map in the other direction. Start with a V-stability \(\mathfrak{s}=\{\mathfrak{s}(X)\}_{X\in \overline{\mathcal{M}}_{g,n}}\) of type \((g,n)\). For any pair \((e;h,A)\in \mathbb{D}_{g,n}\), pick a vine curve \(V=C_1\cup C_2\in \overline{\mathcal{M}}_{g,n}\) such that \(\operatorname{type}(C_1)=(e;h,A)\). Set \(\sigma^\mathfrak{s}(e;h,A):=\mathfrak{s}(V)_{C_1}\) which is well-defined, i.e. it is independent of the chosen \(V\), by Remark 5[R:VStab-Uni1]. Property (1) of [1] for \(\mathfrak{s}(V)\) applied to \(C_1\in \operatorname{BCon}(V)\) and \(C_2=C_1^c\) gives Property [E:condUni1] of Definition 5 for \(\sigma^\mathfrak{s}\). Let us now check that \(\sigma^\mathfrak{s}\) also satisfies Property [E:condUni2] of Definition 5. Fix a triangle \(\Delta=[(e_i;h_i,A_i); i=1,2,3]\) in \(\mathbb{D}_{g,n}\) and consider a triangular curve \((T,p_k)\in \overline{\mathcal{M}}_{g,n}\) in the triangular stratum corresponding to \(\Delta\) as in Remark 7, i.e. \(T=C_1\cup C_2\cup C_3\) with \(C_i\) smooth of genus \(h_i\) with the property that \(A_i=\{k\in [n]\: : p_k\in C_i\}\) and \(t_{i}:=|C_i\cap C_i^c|\). In particular, \(\operatorname{type}(C_i)=(e_i;h_i,A_i)\). For any \(1\leq i\leq 3\), denote by \(\xi_i:V_i\rightsquigarrow T\) the geometric specialization that corresponds to the smoothing of the nodes of \(C_j\cap C_k\) where \(\{1,2,3\}=\{i,j,k\}\). Then \(V_i\) is a vine curve with a component \(\widetilde{C}_i\) such that \((\xi_i)_*(\widetilde{C}_i)=C_i\). By the compatibility of the V-stability conditions of \(\mathfrak{s}\) and the definition of \(\sigma^\mathfrak{s}\), we get that \[\mathfrak{s}(T)_{C_i}=\mathfrak{s}(T)_{(\xi_i)_*(\widetilde{C}_i)}=(\xi_i^*\mathfrak{s}(T))_{\widetilde{C}_i}=\mathfrak{s}(V_i)_{\widetilde{C}_i}=\sigma^\mathfrak{s}(e_i; h_i,A_i).\] Therefore, Property (2) of [1] for \(\mathfrak{s}(T)\) applied to \(T=C_1\cup C_2\cup C_3\) corresponds to Property [E:condUni2] of Definition 5 for \(\sigma^\mathfrak{s}\).

We will now show that the two maps \(\sigma\mapsto \mathfrak{s}^\sigma\) and \(\mathfrak{s}\mapsto \sigma^\mathfrak{s}\) are inverses one of the other. The fact that \(\sigma^{\mathfrak{s}^{\sigma}}=\sigma\) follows immediately from the definitions. In order to prove that \(\mathfrak{s}^{\sigma^\mathfrak{s}}=\mathfrak{s}\), consider a curve \(X\in \overline{\mathcal{M}}_{g,n}\) and any \(Y\in \operatorname{BCon}(X)\). Denote by \(\xi:\widetilde{X}\rightsquigarrow X\) the geometric specialization that corresponds to the smoothing of all the nodes of \(X\) except those belonging to \(Y\cap Y^c\). Then \(\widetilde{X}\) is a vine curve of \(\overline{\mathcal{M}}_{g,n}\) with an irreducible component \(\widetilde{Y}\) such that \(\xi_*(\widetilde{Y})=Y\). Since \(\xi^*\mathfrak{s}(X)=\mathfrak{s}(\widetilde{X})\) by the assumption on \(\mathfrak{s}\), we get that \[\label{E:calco1} \mathfrak{s}(X)_Y=\mathfrak{s}(X)_{\xi_*(\widetilde{Y})}=\xi^*\mathfrak{s}(X)_{\widetilde{Y}}=\mathfrak{s}(\widetilde{X})_{\widetilde{Y}}.\tag{16}\] On the other hand, by the definition of the two maps \(\sigma\mapsto \mathfrak{s}^\sigma\) and \(\mathfrak{s}\mapsto \sigma^\mathfrak{s}\), we have that \[\label{E:calco2} \mathfrak{s}^{\sigma^\mathfrak{s}}(X)_Y=\sigma^\mathfrak{s}(\operatorname{type}_X(Y))=\sigma^\mathfrak{s}(\operatorname{type}_{\widetilde{X}}(\widetilde{Y}))=\mathfrak{s}(\widetilde{X})_{\widetilde{Y}}.\tag{17}\] By comparing 16 and 17 , we get that \(\mathfrak{s}^{\sigma^s}=\mathfrak{s}\), as required.

Finally, it is clear from the definition of the two maps \(\sigma\mapsto \mathfrak{s}^\sigma\) and \(\mathfrak{s}\mapsto \sigma^\mathfrak{s}\) that we have \[\sigma_1\geq \sigma_2\Leftrightarrow \mathfrak{s}^{\sigma_1}(X)\geq \mathfrak{s}^{\sigma_2}(X) \text{ for any } X\in \overline{\mathcal{M}}_{g,n}.\] ◻

Proof of Corollary 2. Part [C:isoH] follows from the fact that \(\overline{J}_{g,n}(\sigma_i)\) is regular (since \(\operatorname{TF}_{g,n}\) is regular and \(\overline{J}_{g,n}(\sigma_i)=\overline{\mathcal{J}}_{g,n}(\sigma_i)\fatslash \operatorname{\mathbb{G}_m}\) because \(\sigma_i\) is general) and that the morphism \(\overline{\mathcal{J}}_{g,n}(\sigma_i)_{\mathbb{C}}\to (\overline{\mathcal{M}}_{g,n})_{\mathbb{C}}\) has full support, which can be deduced arguing as in [36]. Indeed, the result of loc. cit. implies that any classical (in the sense of [1]) relative fine compactified Jacobian over a versal family of nodal curves has full support. However, the fact that the relative compactified Jacobian is classical is only used in the computation of the class in the Grothendieck group of varieties of a classical fine compactified Jacobian of a nodal curve over \(k=\overline{k}\), see [36]. Since the same formula holds true for fine V-compactified Jacobians of nodal curves by [15] and the geometric fibers of \(\overline{J}_{g,n}(\sigma_i)\to \overline{\mathcal{M}}_{g,n}\) are fine V-compactified Jacobians by Theorem 1, the proof of [36] extends to this case.

Part [C:isoD]: since \(\overline{J}_{g,n}^\chi(\sigma_i)\) are fine compactified universal Jacobians (for \(i=1,2\)), then we can pick a tautological sheaf \(\mathcal{I}(\sigma_i):=\mathcal{I}^{\operatorname{taut}}_{g,n}(\sigma_i)\) on the universal family \(\overline{J}_{g,n}^\chi(\sigma_i)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n}\to \overline{J}_{g,n}^\chi(\sigma_i)\) (a fact that it is obvious for \(n\geq 1\) because the universal family has sections and that is proved in Corollary 5 for \(n=0\)). Consider the sheaf \(p_{13}^*(\mathcal{I}(\sigma_1))\otimes p_{23}^*(\mathcal{I}(\sigma_2))\) on \(\overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n}\xrightarrow{\pi} \overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2)\), where \(p_{i3}\) is the projection onto the \(i\)-th and third factor. Over the big open substack \[\iota: (\overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2))^\sharp\hookrightarrow \overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2)\] whose geometric points consist of pairs of sheaves \((I_1,I_2)\) on \(C\in \overline{\mathcal{M}}_{g,n}(k=\overline{k})\) such that for any node \(p\in C\) either \(I_1\) or \(I_2\) is free, the sheaf \(p_{13}^*(\mathcal{I}(\sigma_1))\otimes p_{23}^*(\mathcal{I}(\sigma_2))\) is flat and hence we can consider the Poincaré line bundle \[\mathcal{P}:=d_{\pi}(p_{13}^*(\mathcal{I}(\sigma_1))\otimes p_{23}^*(\mathcal{I}(\sigma_2)))^{-1}\otimes d_{\pi}(p_{13}^*(\mathcal{I}(\sigma_1)))\otimes d_{\pi}(p_{23}^*(\mathcal{I}(\sigma_2)))\in \operatorname{Pic}((\overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2))^\sharp).\] The pushforward \(\overline{\mathcal{P}}:=\iota_*(\mathcal{P})\) (called the Poincaré or Arinkin sheaf) is a maximal Cohen-Macaulay sheaf that is flat over each factor (see [37], [38], [29]). The sheaf \(\overline{\mathcal{P}}\) satisfies the following identity in the derived category of \(\overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_1)\) \[\label{E:convP} Rp_{\pi_{13*}}(\pi_{12}^*((\overline{\mathcal{P}})^\vee)\otimes \pi_{23}^*(\overline{\mathcal{P}}))\cong \mathcal{O}_{\Delta}[-g],\tag{18}\] where \(\pi_{ij}\) denotes the projection onto the \(i\)-th and \(j\)-th term of the product \(\overline{J}_{g,n}^\chi(\sigma_1)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_2)\times_{\overline{\mathcal{M}}_{g,n}}\overline{J}_{g,n}^\chi(\sigma_1)\). The identity 18 can be checked étale-locally on the base and hence it follows from [38] where it is proved for two relative classical compactified Jacobians over the semiuniversal family of a reduced curve with planar singularity; however, an inspection of the proof reveals that the same arguments work for two relative V-compactified Jacobians. The identity 18 now implies that the Fourier-Mukai transform with kernel \(\overline{\mathcal{P}}\) gives the desired equivalence of triangulated categories. ◻

Proof. This is immediate from Formula 24 . ◻

Proof. Part [T:cla-cJUni1] follows by combining Lemma 1 and Theorem 6.

Let us prove Part [T:cla-cJUni2]. First of all, since the arrangement of hyperplanes \(\mathcal{A}_{g,n}^\chi\) is rational, we can perturb \(L\in \operatorname{PicRel}^{\chi}(\mathbb{R})\) to a rational relative line bundle in such a way that the perturbed line bundle lies inside the same region. In other words, we can assume that \(L\) is a rational relative line bundle on \(\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}\). Then we can write \[L=\frac{M}{r} \text{ for some } r\in \mathbb{N}_{>0} \text{ and } M\in \operatorname{PicRel}(\mathbb{Z}) \text{ with } \deg_{\pi}(M)=r\chi.\] Consider now the following vector bundle on \(\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}\) \[E=\mathcal{O}_{\overline{\mathcal{C}}_{g,n}}^{\oplus(r-1)}\oplus M^{-1},\] which has relative slope equal to \(-\chi\). By construction, we have that \(\underline\deg(L)=\psi(E)\in \operatorname{Pol}_{g,n}^\chi\), where \(\psi(E)\) is the Esteves [12] relative numerical polarization on \(\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}\) (see [1]. This implies (using the notation of [1]) that \[\overline{\mathcal{J}}_{g,n}([L])= \overline{\mathcal{J}}_{\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}}(E) \text{ and } \overline{J}_{g,n}([L])= \overline{J}_{\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}}(E).\] Then we conclude by applying [1] to an étale cover of \(\overline{\mathcal{M}}_{g,n}\) by an algebraic space. ◻

Proof. This is straightforward using Definition 3. ◻

Proof. The action of \(\widetilde{\operatorname{PR}}_{g,n}\) on \(\operatorname{TF}_{g,n}\) is well-defined: \(L\cdot \mathcal{I}\) is well-defined since \(L\) is a relative line bundle on \(\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}\), \(\iota\cdot \mathcal{I}\) is well-defined as observed in [3] and clearly we have that \(\iota\cdot (L\cdot \mathcal{I})=L^{-1}\cdot (\iota\cdot \mathcal{I})\).

In order to prove the second statement, we have to prove that for any \(\mathfrak{s}\in \operatorname{VStab}_{g,n}\) we have that \[L\cdot \overline{\mathcal{J}}_{g,n}(\mathfrak{s})\subseteq \overline{\mathcal{J}}_{g,n}(L\cdot \mathfrak{s}) \text{ and } \iota\cdot \overline{\mathcal{J}}_{g,n}(\mathfrak{s})\subseteq \overline{\mathcal{J}}_{g,n}(\iota\cdot \mathfrak{s}),\] for any \(L\in \operatorname{PicRel}_{g,n}(\mathbb{Z})\). This amounts to proving that, for any given \(X\in \overline{\mathcal{M}}_{g,n}\) and any \(I\in \operatorname{TF}_X\) which is \(\mathfrak{s}(X)\)-semistable, we have that

(i) \(I\otimes L_{|X}\) is \((L\cdot \mathfrak{s})(X)\)-semistable, for any \(L\in \operatorname{PicRel}_{g,n}(\mathbb{Z})\);

(ii) \(I^*={\mathcal{H}om}(I,\omega_X)\) is \((\iota \cdot \mathfrak{s})(X)\)-semistable.

Property [E:ss-L] follows from the definition of \((L\cdot \mathfrak{s})(X)\) (see Lemma-Definition 14) together with the fact that \[\chi((I\otimes L_{|X})_Y)=\chi(I_Y)+\deg_Y(L_{|X}) \text{ for any } Y\in \operatorname{BCon}(X).\]

Property [E:ss-iota] follows from the following computation for any \(Y\in \operatorname{BCon}(X)\): \[\begin{align} & \chi((I^*)_Y)=\chi((I_Y)^*)+|Y\cap Y^c\cap \operatorname{NF}(I)^c| & \text{ by Lemma \ref{L:duality}}\\ & =-\chi(I_Y)+|Y\cap Y^c\cap \operatorname{NF}(I)^c| & \text{ by Serre duality,}\\ & = \chi(I_{Y^c})-\chi(I) & \text{ by \eqref{E:2restY} and \eqref{E:add-chi},}\\ & \geq \mathfrak{s}(X)_{Y^c}-|\mathfrak{s}(X)| & \text{ since I is \mathfrak{s}(X)-semistable,}\\ &=\begin{cases} -\mathfrak{s}(X)_Y & \text{ if } Y\in \mathcal{D}(\mathfrak{s}(X)),\\ -s(X)_Y+1 & \text{ if } Y\not \in \mathcal{D}(\mathfrak{s}(X)). \end{cases} & \text{ by Definition \ref{D:VStabX},}\\ & =(\iota\cdot \mathfrak{s}(X))_Y & \text{ by Lemma-Definition \ref{LD:PR-VStab}.} \end{align}\] ◻

Proof. Equality 28 follows from 27 by taking the Euler characteristic.

In order to prove 27 , consider the two exact sequences \[\label{E:2seqex} \begin{align} & 0 \to \omega_Y\xrightarrow{a} \omega_X\xrightarrow{b} \omega_{Y^c}(Y\cap Y^c)\to 0,\\ & 0 \to I_{Y^c}(-Y\cap Y^c\cap \operatorname{NF}(I)^c)\xrightarrow{\beta} I \xrightarrow{\alpha} I_Y\to 0, \end{align}\tag{29}\] where the first exact sequence follows by adjunction and the second exact sequence follows from 5 . From a straightforward diagram chasing with the two exact sequences in 29 , we get following exact sequence \[\label{E:leftex} \begin{align} 0 \to {\mathcal{H}om}(I_Y,\omega_Y) \rightarrow & {\mathcal{H}om}(I,\omega_X) \xrightarrow{} {\mathcal{H}om}(I_{Y^c}(-Y\cap Y^c\cap \operatorname{NF}(I)^c), \omega_X)\oplus {\mathcal{H}om}(I,\omega_{Y^c}(Y\cap Y_c))\\ f \mapsto & a\circ f\circ \alpha & \\ &g \mapsto (g\circ \beta,b\circ g)\\ \end{align}\tag{30}\] This shows that \((I_Y)^*\) is a subsheaf of \(I^*\) (supported on \(Y\)) such that the quotient \(I^*/(I_Y)^*\) is a torsion-free sheaf supported on \(Y^c\), and hence that it is the largest subsheaf of \(I^*\) supported on \(Y\), which implies that \((I_Y)^*=\leftindex_{Y}{(I^*)}\). Combining this with 5 , we deduce that \[(I_Y)^*=\leftindex_{Y}{(I^*)}=(I^*)_Y(-Y\cap Y^c\cap \operatorname{NF}(I)^c),\] as required. ◻

Proof. The implication \(\Longleftarrow\) follows from Proposition 16 and Theorem 6.

In order to show the other implication, fix a birational morphism \(\phi:\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\rightarrow\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_2)\) over \(\overline{\mathcal{M}}_{g,n}\). The proof consists of two steps.

Step I: There exists an element \(g\in \widetilde{\operatorname{PR}}_{g,n}\) such that the composition \[\psi: \overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\xrightarrow{\phi}\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_2)\xrightarrow[\cong]{\cdot g}\overline{\mathcal{J}}_{g,n}(g\cdot\mathfrak{s}_2)\] restricts to an inclusion \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)^{\leq 1}\subseteq \overline{\mathcal{J}}_{g,n}(g\cdot \mathfrak{s}_2)^{\leq 1}\) over the open substack \(\overline{\mathcal{M}}_{g,n}^{\leq 1}\) parametrizing stable curves with at most one node, where we set \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s})^{\leq 1}:=\overline{\mathcal{J}}_{g,n}(\mathfrak{s})_{|\overline{\mathcal{M}}_{g,n}^{\leq 1}}\)

Indeed, by post-composing with the multiplication by an element of the form \((\beta\omega_{\pi}+\sum_{i=1}^n \alpha_i\Sigma_i,\iota^{e})\in \widetilde{\operatorname{PR}}_{g,n}\) (with \(e=0,1\)), we can assume that the morphism becomes the identity on the universal Jacobian over smooth curves \(\mathcal{J}_{g,n}\) (using that any automorphism of \(\mathcal{J}_{g,n}\) over \(\mathcal{M}_{g,n}\) is the multiplication by one such element, see [3]). Then, by further post-componing with an element of the form \(\sum_{(h,A)\in \mathbb{B}_{g,n}}\gamma_{(h,A)}\mathcal{O}(C(h,A))\in \operatorname{PicRel}_{g,n}(\mathbb{Z})\), we can assume that the morphism becomes an inclusion over the generic point of each boundary divisor of \(\overline{\mathcal{M}}_{g,n}\) (using the structure of compactified Jacobian stacks of nodal curves with at most one node, see e.g. [2]).

Step II: The morphism \(\psi\) is the inclusion \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\subseteq \overline{\mathcal{J}}_{g,n}(g\cdot \mathfrak{s}_2)\) (and hence \(\mathfrak{s}_1\geq g\cdot \mathfrak{s}_2\) by Theorem 6).

Indeed, by Proposition 16 and [1] it is enough to show that \(\psi\) commutes with the open embeddings of the domain and of the codomain in \(\operatorname{TF}_{g,n}\). In order to show this, consider the universal sheaf \(\mathcal{I}_1\) (resp. \(\mathcal{I}_2\)) on \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n}\) (resp. on \(\overline{\mathcal{J}}_{g,n}(g\cdot \mathfrak{s}_2)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n}\)), which is the restriction from \(\operatorname{TF}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}}\overline{\mathcal{C}}_{g,n}\) of the universal sheaf. The fact that \(\psi\) commutes with the open embeddings of the domain and of the codomain in \(\operatorname{TF}_{g,n}\) is equivalent to the equality of sheaves \[\label{E:eq-shea} (\psi\times \operatorname{id})^*(\mathcal{I}_2)=\mathcal{I}_1.\tag{31}\] By Step I, we have that \[\label{E:eq-open} (\psi\times \operatorname{id})^*(\mathcal{I}_2)_{|(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n})^{\leq 1}}=(\mathcal{I}_1)_{|(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n})^{\leq 1}},\tag{32}\] where \((\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n})^{\leq 1}=\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)^{\leq 1}\times_{\overline{\mathcal{M}}_{g,n}^{\leq 1}} \overline{\mathcal{C}}_{g,n}^{\leq 1}\) is the open substack obtained by restricting to \(\overline{\mathcal{M}}_{g,n}^{\leq 1}\). Now, the equality 32 implies the equality 31 by [3], using that \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\subset \operatorname{TF}_{g,n}\) is regular, that \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)\times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{C}}_{g,n}\subset \operatorname{TF}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}}\overline{\mathcal{C}}_{g,n}\) is Cohen-Macaulay and regular in codimension one (by the same proof of [3]) and that \(\overline{\mathcal{J}}_{g,n}(\mathfrak{s}_1)^{\leq 1}\subset \overline{\mathcal{J}}_{g,n}\) is a big open subset (i.e. its complement has codimension at least two). ◻

Proof. We will use throughout this proof the \(\widetilde{\operatorname{PR}}_{g,n}\)-equivariant isomorphism of poset \(\Sigma_{g,n}\cong \operatorname{VStab}_{g,n}\) of Proposition 10. We will prove a chain of implications.

\(\eqref{C:iso-UniSt1}\Rightarrow \eqref{C:iso-UniSt2}\) follows from Proposition 16.

\(\eqref{C:iso-UniSt2}\Rightarrow \eqref{C:iso-UniSt3}\): we have only to show that if \(\overline{\mathcal{J}}_{g,n}(\sigma_1)\) and \(\overline{\mathcal{J}}_{g,n}(\sigma_2)\) are isomorphic over \(\overline{\mathcal{M}}_{g,n}\) then \(\mathcal{D}(\sigma_1)=\mathcal{D}(\sigma_2)\). This follows from the fact that \((e;h,A)\in \mathcal{D}(\sigma_i)\) if and only if the fiber of \(\overline{\mathcal{J}}_{g,n}(\sigma_i)\) over the geometric generic point of the vine stratum corresponding to \((e;h,A)\) (see Remark 7) is non-fine, which indeed occurs if and only if this fiber has \((e+1)\)-irreducible components (see [10]).

\(\eqref{C:iso-UniSt3}\Rightarrow \eqref{C:iso-UniSt1}\): the existence of the birational morphism \(\phi\) as in [C:iso-UniSt3] implies, by Theorem 18, that there exists \(g\in \widetilde{\operatorname{PR}}_{g,n}\) such that \(\sigma_1\geq g\cdot \sigma_2\). Since we have that \(\mathcal{D}(\sigma_1)=\mathcal{D}(\sigma_2)=\mathcal{D}(g\cdot \sigma_2)\) by assumption and Proposition 33[P:Deg-map1], Remark 9 implies that \(\sigma_1=g\cdot \sigma_2\). ◻

Proof. This follows immediately from the fact that all the triangles of \(\mathbb{D}_{g,n}\) are contained in \({^{ns}\mathbb{D}_{g,n}}\). ◻

Proof. This follows from the fact that \[X\in \overline{\mathcal{M}}_{g,n}^{ns}\Leftrightarrow \operatorname{type}_X(Y)\in {^{ns}\mathbb{D}_{g,n}}\text{ for any } Y\in \operatorname{BCon}(X),\] together with [1]. ◻

Proof. The proof is straightforward from Remark 15. ◻

Proof. Using Lemma 5[L:Sigma-ac1], it is enough to prove that the action of \(\operatorname{PR}_{g,n}\) on \(\operatorname{^{ns}\Sigma}_{g,n}\) has finitely many orbits. Moreover, Lemma 5[L:Sigma-ac2] implies that, by translating by an element of \(\operatorname{PicRel}^{op}_{g,n}(\mathbb{Z})\), any element of \(\operatorname{^{ns}\Sigma}_{g,n}\) lies in the same orbit of a unique normalized element \(\sigma\in \operatorname{^{ns}\Sigma}_{g,n}\), i.e. an element \(\sigma\in \operatorname{^{ns}\Sigma}_{g,n}\) such that \[\label{E:norm-sigma} { \left\{\begin{align} & \sigma(3;0,\emptyset)=0, \\ & \sigma(2;0,\{i\})=0 \text{ for any } i\in [n]. \\ \end{align}\right.}\tag{37}\] We now show that there are a finite number of normalized elements.

By applying Remark 8 to the triangle \(\Delta=[(3;0,\emptyset),(e+1;h-1,A), (e; g-h-e+1,A^c)]\) of \({^{ns}\mathbb{D}_{g,n}}\) for any \(h\geq 1\) and using the first vanishing in 37 , we get that \[\label{E:condiz1} -1\leq \sigma(e;h,A)-\sigma(e+1;h-1,A)\leq 0 \text{ for any } (e;h,A)\in {^{ns}\mathbb{D}_{g,n}}\text{ such that } h\geq 1.\tag{38}\]

By applying Remark 8 to the triangle \(\Delta=[(3;0,\emptyset),(e-1;h,A), (e; g-h-e+1,A^c)]\) of \({^{ns}\mathbb{D}_{g,n}}\) for any \(e\geq 3\) and using the first vanishing in 37 , we get that \[\label{E:condiz2} -1\leq \sigma(e;h,A)-\sigma(e-1;h,A)\leq 0 \text{ for any } (e;h,A)\in {^{ns}\mathbb{D}_{g,n}}\text{ such that } e\geq 3.\tag{39}\]

By applying Remark 8 to the triangle \(\Delta=[(2;0,\{i\}),(e;h,A-\{i\}), (e; g-h-e+1,A^c)]\) of \({^{ns}\mathbb{D}_{g,n}}\) for any \(i\in A\) and using the second vanishing in 37 , we get that \[\label{E:condiz3} -1\leq \sigma(e;h,A)-\sigma(e;h,A-\{i\})\leq 0 \text{ for any } (e;h,A)\in {^{ns}\mathbb{D}_{g,n}}\text{ such that } i\in A.\tag{40}\]

Now, by iterating 38 \(h\)-times, we get that \[\label{E:condiz1bis} -h\leq \sigma(e;h,A)-\sigma(e+h;0,A)\leq 0 \text{ for any } (e;h,A)\in {^{ns}\mathbb{D}_{g,n}}.\tag{41}\] By iterating 39 and using the first vanishing in 37 , we get that \[\label{E:condiz2bis} { \left\{\begin{align} & -(e-2)\leq \sigma(e;0,A)-\sigma(2;0,A)\leq 0 & \text{ for any } e\geq 2 \text{ and any } |A|>0;\\ & -(e-3)\leq \sigma(e;0,\emptyset)-\sigma(3;0,\emptyset)=\sigma(e;0,\emptyset)\leq 0 & \text{ for any } e\geq 3.\\ \end{align}\right.}\tag{42}\] By iterating 40 and using the second vanishing in 37 , we get that \[\label{E:condiz3bis} -(|A|-1)\leq \sigma(2;0,A)-\sigma(2;0,\{i\})=\sigma(2;0,A)\leq 0 \: \text{ if } i\in A.\tag{43}\]

By putting together 41 , 42 and 43 , we get that \[\label{E:final-s} -(2h+e+|A|-3)\leq \sigma(e;h,A)\leq 0 \text{ for any } (e;h,A)\in {^{ns}\mathbb{D}_{g,n}},\tag{44}\] which show that the set of normalized elements of \({^{ns}\mathbb{D}_{g,n}}\) is finite, as we wanted to show. ◻

Proof. The implications \(\eqref{T:iso-UniSp2}\Rightarrow \eqref{T:iso-UniSp4}\) and \(\eqref{T:iso-UniSp3}\Rightarrow \eqref{T:iso-UniSp4}\) are obvious.

The implication \(\eqref{T:iso-UniSp1}\Rightarrow \eqref{T:iso-UniSp2}\) follows by combining Proposition 16 and Lemmas 4 and 5[L:Sigma-ac1].

We now prove the implication \(\eqref{T:iso-UniSp1}\Rightarrow \eqref{T:iso-UniSp3}\). Up to replacing \(\sigma_2\) with a conjugate for the action of \(\widetilde{\operatorname{PR}}_{g,n}\), we can assume that \(\sigma_1^{ns}=\sigma_2^{ns}\). In particular, we have that \(|\sigma_1|=|\sigma_2|:=\chi\). By Lemma 3 and Proposition 34[P:Dsep4], we can choose \(\overline{\sigma}_i\in \Sigma_{g,n}^\chi\) (for \(i=1,2\)) such that \[\overline{\sigma}_i\leq \sigma_i, \quad (\overline{\sigma}_i)^{ns}= \sigma_i^{ns} \text{ and } \operatorname{^{s}\mathcal{D}}(\overline{\sigma}_i^s) \text{ is the minimum element of } \operatorname{Im}\operatorname{^{s}\mathcal{D}}^{\chi}.\] By Lemma 5[L:Sigma-ac1] and the assumption \(\sigma_1^{ns}=\sigma_2^{ns}\), we deduce that \(\overline{\sigma}_1\) and \(\overline{\sigma}_2\) are in the same orbit for the action of \(\widetilde{\operatorname{PR}}_{g,n}\) on \(\Sigma_{g,n}\). Hence, by applying Proposition 16, we deduce that the universal compactified Jacobian stacks \(\overline{\mathcal{J}}_{g,n}(\overline{\sigma}_1)\) and \(\overline{\mathcal{J}}_{g,n}(\overline{\sigma}_2)\) are isomorphic over \(\overline{\mathcal{M}}_{g,n}\), which implies, by passing to their relative good moduli spaces, that \[\overline{J}_{g,n}(\overline{\sigma}_1) \text{ and } \overline{J}_{g,n}(\overline{\sigma}_2) \text{ are isomorphic over } \overline{\mathcal{M}}_{g,n}.\] In order to conclude the proof of this implication, it remains to prove that (for \(i=1,2\)) the inclusion \(\overline{\mathcal{J}}_{g,n}(\sigma_i)\subseteq \overline{\mathcal{J}}_{g,n}(\overline{\sigma}_i)\) (see Theorem 1) induces, by passing to their relative good moduli spaces, an isomorphism \[\Phi: \overline{J}_{g,n}(\sigma_i)\xrightarrow{\cong} \overline{J}_{g,n}(\overline{\sigma}_i).\] This can be proved as follows. First, the fact that \(\Phi\) is an isomorphism can be checked on geometric fibers by [40] and [41], using that \(\overline{J}_{g,n}(\overline{\sigma}_i)\) (as well as \(\overline{J}_{g,n}(\sigma_i)\)) is flat over \(\overline{\mathcal{M}}_{g,n}\) by [1]. We then conclude using the assumptions \(\overline{\sigma}_i\leq \sigma_i\) and \((\overline{\sigma}_i)^{ns}= \sigma_i^{ns}\) and Lemma 6 below.

We finally prove the implication \(\eqref{T:iso-UniSp4}\Rightarrow \eqref{T:iso-UniSp1}\), which will conclude the proof. By assumption, we have an isomorphism over \(\overline{\mathcal{M}}_{g,n}^{ns}\) \[\label{E:iso-Psi} \Psi:\overline{J}_{g,n}(\sigma_1)_{|\overline{\mathcal{M}}_{g,n}^{ns}}\xrightarrow{\cong}\overline{J}_{g,n}(\sigma_2)_{|\overline{\mathcal{M}}_{g,n}^{ns}}.\tag{45}\] By [3], the restriction of \(\Psi\) over \({\mathcal{M}}_{g,n}\) is given by the action of a uniquely defined element of \(g\in \operatorname{PR}_{g,n}\). More precisely, there exists a unique element \(g\in \operatorname{PR}_{g,n}\) such that, for any lift \(\widetilde{g}\in \widetilde{\operatorname{PR}}_{g,n}\) of \(g\), we have that \[\Psi_{|{\mathcal{M}}_{g,n}}:\overline{J}_{g,n}(\sigma_1)_{|{\mathcal{M}}_{g,n}}=\operatorname{TF}_{{\mathcal{C}}_{g,n}/{\mathcal{M}}_{g,n}}^\chi\fatslash \operatorname{\mathbb{G}_m}\xrightarrow[\cong]{\cdot \widetilde{g}}\operatorname{TF}_{{\mathcal{C}}_{g,n}/{\mathcal{M}}_{g,n}}^\chi\fatslash \operatorname{\mathbb{G}_m}=\overline{J}_{g,n}(\widetilde{g}\cdot \sigma_1)_{|{\mathcal{M}}_{g,n}}\] where the action of \(\widetilde{g}\) is the one in Proposition 16. Therefore, up to replacing \(\sigma_1\) with \(\widetilde{g}\cdot \sigma_1\) (and hence \(\sigma_1^{ns}\) with \(g\cdot \sigma_1^{ns}\)) and \(\Psi\) with \(\Psi\circ (\cdot \widetilde{g})^{-1}\), we can (and will) assume that \[\label{E:Psi-ass} \Psi_{|{\mathcal{M}}_{g,n}}=\operatorname{id}.\tag{46}\] We now show that this implies the equality \(\sigma_1^{ns}=\sigma_2^{ns}\), which will conclude the proof. Fix \((e;h,A)\in {^{ns}\mathbb{D}_{g,n}}\) (so that \(e\geq 2\)) and pick a vine curve \(X=C_1\cup C_2\in \overline{\mathcal{M}}_{g,n}\) such that \(\operatorname{type}_X(C_1)=(e;h,A)\) and \(\operatorname{type}_X(C_2)=(e;h,A)^c\). For \(i=1,2\), consider the V-stability \(\mathfrak{s}_i:=\sigma^{\sigma_i}(X)\) on \(X\) as in Proposition 10, which satisfies \((\mathfrak{s}_i)_{C_1}=\sigma_i(e;h,A)\) and \((\mathfrak{s}_i)_{C_2}=\sigma_i((e;h,A)^c)\). It is then enough to show that \(\mathfrak{s}_1=\mathfrak{s}_2\).

For that purpose, consider the effective semiuniversal deformation space \(\Phi:\mathfrak{X}\to \operatorname{Spec}R_X\) of \(X\), as in [1]. Since \(R_X\) is the complete local ring of \(\overline{\mathcal{M}}_{g,n}^{ns}\subset \overline{\mathcal{M}}_{g,n}\) at \(X\), there is a morphism \(\eta:\operatorname{Spec}R_X\to \overline{\mathcal{M}}_{g,n}^{ns}\) sending \(o\) into \(X\). By pulling back the isomorphism \(\Psi\) of 45 along \(\eta\) and using that good moduli spaces are functorial, we get an isomorphism \[\Psi_X:\overline{J}_{\mathfrak{X}/R_X}(\mathfrak{s}_1)\xrightarrow{\cong} \overline{J}_{\mathfrak{X}/R_X}(\mathfrak{s}_1).\] Moreover, since the open subset \(U_X\) of \(\operatorname{Spec}R_X\) parametrizing smooth fibers of \(\mathfrak{X}\to \operatorname{Spec}R_X\) is the pull-back of \({\mathcal{M}}_{g,n}\) via \(\eta\), the assumption from 46 translates into the fact that \[(\Psi_X)_{|U_X}=\operatorname{id}.\] Now we can apply Lemma 7 below in order to conclude that \(\mathfrak{s}_1=\mathfrak{s}_2\), and we are done. ◻

Proof. Since the biconnected subcurves \(Y\subset X\) such that \(|Y\cap Y^c|=1\) do not appear in Condition [D:VStabX2] of Definition 1, we can pick a V-stability condition \(\widehat\mathfrak{s}\in \operatorname{VStab}^\chi(X)\) such that \[\label{E:whs} { \left\{\begin{align} & \widehat\mathfrak{s}\leq \overline{\mathfrak{s}}\leq \mathfrak{s},\\ & \widehat\mathfrak{s}_Y=\overline{\mathfrak{s}}_Y=\mathfrak{s}_Y \text{ for any } Y\in \operatorname{BCon}(X) \text{ such that } |Y\cap Y^c|\geq 2, \\ & \mathcal{D}(\widehat\mathfrak{s})\supseteq \{Y\in \operatorname{BCon}(X): \: |Y\cap Y^c|=1\}. \end{align}\right.}\tag{47}\] Then we have inclusions \(\overline{\mathcal{J}}_X(\widehat\mathfrak{s})\subseteq \overline{\mathcal{J}}_X(\mathfrak{s})\subseteq \overline{\mathcal{J}}_X(\overline{\mathfrak{s}})\), that induce the following maps by passing to the good moduli spaces \[\Upsilon_1: \overline{\mathcal{J}}_X(\mathfrak{s})\xrightarrow{\Upsilon} \overline{\mathcal{J}}_X(\overline{\mathfrak{s}})\xrightarrow{\Upsilon_2} \overline{\mathcal{J}}_X(\widehat\mathfrak{s}).\] In order to show that \(\Upsilon\) is an isomorphism, it is enough to show that \(\Upsilon_1\) and \(\Upsilon_2\) are isomorphisms. Let us show that \(\Upsilon_1\) is an isomorphism, the proof for \(\Upsilon_2\) being the same.

Denote by \(\{Z_i\}\) the closure of the connected components of \(X-X_{\rm sep}\), where \(X_{\rm sep}\) is the set of separating nodes of \(X\). Then we get a decomposition into connected subcurves \[X=\bigcup_i Z_i \text{ with } Z_i\wedge Z_j=\emptyset \text{ and } Z_i\cap Z_j\subset X_{\rm sep} \text{ for any } i\neq j.\] Because of the third property in 47 , we have that \(Z_i \in \widehat\mathcal{D}(\widehat\mathfrak{s})\). Consider the induced V-stabilities \(\widehat\mathfrak{s}(Z_i)\) as in [1]. By [1], we have a closed embedding \[\begin{align} \Xi: \prod_i \overline{\mathcal{J}}_{Z_i}(\widehat s(Z_i))& \hookrightarrow \overline{\mathcal{J}}_X(\widehat\mathfrak{s})\\ (I_i) & \mapsto \bigoplus_i I_i. \end{align}\] Since the subcurves \(Z_i\) meet only at separating nodes of \(X\), by [2] we deduce that any sheaf of \(\overline{\mathcal{J}}_X(\widehat\mathfrak{s})\) (and hence of \(\overline{\mathcal{J}}_X(\mathfrak{s})\)) isotrivially specializes to a unique sheaf of \(\prod_i \overline{\mathcal{J}}_{Z_i}(\widehat s(Z_i))\). This implies, using the properties [42] of good moduli spaces, that the map \(\Xi\) induces, by passing to the good moduli spaces, two isomorphisms \[{ \left\{\begin{align} & |\Xi|:\prod_i \overline{J}_{Z_i}(\widehat\mathfrak{s}(Z_i)) \xrightarrow{\cong} \overline{J}_X(\widehat\mathfrak{s}),\\ & \overline{J}_X(\mathfrak{s}) \xrightarrow{\Upsilon_1} \overline{J}_X(\widehat\mathfrak{s}) \xrightarrow[\cong]{|\Xi|^{-1}} \prod_i \overline{J}_{Z_i}(\widehat\mathfrak{s}(Z_i)). \end{align}\right.}\] We conclude that \(\Upsilon_1\) is an isomorphism, as required. ◻

Proof. Denote by \(\Delta_X\) the closed subset of \(\operatorname{Spec}R_X\) parametrizing locally trivial deformations of \(X\). Observe that \(\Delta_X\) has codimension \(e\geq 2\) in \(\operatorname{Spec}R_X\) and its complement \(V_X:=\operatorname{Spec}R_X- \Delta_X\) is the open subset of \(\operatorname{Spec}R_X\) parametrizing irreducible fibers of \(\mathfrak{X}\to \operatorname{Spec}R_X\). Therefore, we also have an isomorphism \[\overline{\operatorname{id}}_{V_X}:\overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)_{|V_X}\xrightarrow{\cong} \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_2)_{|V_X}\] induced by the identification \(\operatorname{id}_{V_X}:\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)_{|V_X}=(\operatorname{TF}_{\mathfrak{X}/R_X}^\chi)_{|V_X}= \overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_2)_{|V_X}\). Since \(\Psi_{|U_X}=\operatorname{id}_{U_X}\) by assumption and \(U_X\) is dense in \(V_X\), we must have that \(\Psi_{|V_X}=\overline{\operatorname{id}}_{V_X}\) using that \(\overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)\to \operatorname{Spec}R_X\) (for \(i=1,2\)) is proper with geometrically reduced fibers by [1].

For \(i=1,2\), denote by \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\) the open subset of \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)\) parametrizing families of \(\overline{\mathfrak{s}}_i\)-stable sheaves, which clearly contains \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)_{|V_X}\). The restriction of \(\Xi_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)\) to \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\) is the \(\operatorname{\mathbb{G}_m}\)-rigidification onto its image \[\overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}:=\Xi_{\mathfrak{X}/R_X}(\overline{s}_i)(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}})\subseteq \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i).\] Moreover, since \(\mathfrak{X}\to \operatorname{Spec}R_X\) admits sections, the \(\operatorname{\mathbb{G}_m}\)-gerbe \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\to \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\) is trivial, i.e. we have a (non-canonical) isomorphism \[\label{E:st-gerbe} \overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\cong \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\times B \operatorname{\mathbb{G}_m}.\tag{48}\]

From the explicit structure of compactified Jacobian spaces of vine curves (see [10]), it follows that the complement of the stable locus \(\overline{J}_X(\mathfrak{s}_i)-\overline{J}_X(\mathfrak{s}_i)^{\operatorname{st}}\) (for \(i=1,2\)) is non-empty if and only if the intersection of all irreducible components of \(\overline{J}_X(\mathfrak{s}_i)\) is irreducible for \(e\geq 3\) (resp. if and only if the singular locus of \(\overline{J}_X(\mathfrak{s}_i)\) is irreducible for \(e=2\)), in which case this irreducible locus is exactly \(\overline{J}_X(\mathfrak{s}_i)-\overline{J}_X(\mathfrak{s}_i)^{\operatorname{st}}\). Therefore, the isomorphism \(\Phi\) restricts to an isomorphism \[\label{E:Psi-st} \Psi^{\operatorname{st}}: \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}\xrightarrow{\cong} \overline{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_2)^{\operatorname{st}}.\tag{49}\] Using the isomorphism 48 and the assumption that \(\Psi^{\operatorname{st}}_{|V_X}=\Psi_{|V_X}=\operatorname{id}_{V_X}\), we can lift \(\Psi^{\operatorname{st}}\) to an isomorphism \[\label{E:Psi-wt} \widetilde{\Psi}: \overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}\xrightarrow{\cong} \overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_2)^{\operatorname{st}} \text{ such that } \widetilde{\Psi}_{|V_X}=\operatorname{id}_{V_X}.\tag{50}\] Denote by \(\mathcal{I}_i\) the universal sheaf on \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_i)^{\operatorname{st}}\times_{\operatorname{Spec}R_X} \mathfrak{X}\) for \(i=1,2\), and consider the two sheaves \(\mathcal{I}_1\) and \(\widetilde{\mathcal{I}}:=(\widetilde{\Psi}\times \operatorname{id}_{\mathfrak{X}})^*(\mathcal{I}_2)\) on \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}\times_{\operatorname{Spec}R_X} \mathfrak{X}\). Since \(\widetilde{\Psi}_{|V_X}=\operatorname{id}_{V_X}\), the two sheaves \(\mathcal{I}_1\) and \((\widetilde{\Psi}\times \operatorname{id}_{\mathfrak{X}})^*(\mathcal{I}_2)\) coincide on the open subset \(\overline{(}\mathcal{J}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}\times_{\operatorname{Spec}R_X} \mathfrak{X})_{|V_X}\) whose complement has codimension \(e\geq 2\). Hence, using that \(\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}\times_{\operatorname{Spec}R_X} \mathfrak{X}\) is Cohen-Macaualy and regular in codimension one (which is proved as in [3]), we can apply [3] in order to conclude that \(\mathcal{I}_1= (\widetilde{\Psi}\times \operatorname{id}_{\mathfrak{X}})^*(\mathcal{I}_2)\). This implies that \(\widetilde{\Psi}=\operatorname{id}\), which then gives \[\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_1)^{\operatorname{st}}=\overline{\mathcal{J}}_{\mathfrak{X}/R_X}(\overline{\mathfrak{s}}_2)^{\operatorname{st}}\subset \operatorname{TF}_{\mathfrak{X}/R_X}^\chi.\] Taking the central fiber, we get that \[\overline{\mathcal{J}}_{X}(\mathfrak{s}_1)^{\operatorname{st}}=\overline{\mathcal{J}}_{X}(\mathfrak{s}_2)^{\operatorname{st}}\subset \operatorname{TF}_X^{\chi},\] which then implies that \(\mathfrak{s}_1=\mathfrak{s}_2\), by using that a line bundle \(L\) on \(X\) belongs to \(\overline{\mathcal{J}}_{X}(\mathfrak{s}_i)^{\operatorname{st}}\) if and only if (for \(1\leq i,j\leq 2\)) \[\chi(L_{C_j})\in { \left\{\begin{align} [(\mathfrak{s}_i)_{C_j},(\mathfrak{s}_i)_{C_j}+e-1] & \text{ if } \mathfrak{s}_i \text{ is general,}\\ [(\mathfrak{s}_i)_{C_j}+1,(\mathfrak{s}_i)_{C_j}+e-1] & \text{ if } \mathfrak{s}_i \text{ is not general.} \end{align}\right.}\] ◻

Proof. Part [T:univ-TF1]: it is enough to show that there exist open substacks \(\mathcal{U}_{\geq 0}\), \(\mathcal{U}_{>0}\), \(\mathcal{V}_{\geq \chi}\) and \(\mathcal{V}_{>\chi}\) of \(\operatorname{TF}_{g,n+1}^\chi\) whose geometric points are given by \[{ \left\{\begin{align} &\mathcal{U}_{\geq 0}(k):=\{(C,I)\in \operatorname{TF}_{g,n+1}^\chi(k)\: : \chi(I_E)\geq 0 \text{ for any exceptional component } E \text{ of } C\},\\ &\mathcal{U}_{>0}(k):=\{(C,I)\in \operatorname{TF}_{g,n+1}^\chi(k)\: : \chi(I_E)> 0 \text{ for any exceptional component } E \text{ of } C\},\\ & \mathcal{V}_{\geq \chi}(k):=\{(C,I)\in \operatorname{TF}_{g,n+1}^\chi(k)\: : \chi(I_{E^c})\geq \chi \text{ for any exceptional component } E \text{ of } C\},\\ & \mathcal{V}_{> \chi}(k):=\{(C,I)\in \operatorname{TF}_{g,n+1}^\chi(k)\: : \chi(I_{E^c})> \chi \text{ for any exceptional component } E \text{ of } C\}.\\ \end{align}\right.}\] In order to show this, consider the universal stabilization morphism \[\label{E:univ-stab} \begin{tikzcd} \mathcal{C}_1:=\overline{\mathcal{C}}_{g,n+1}\times_{\overline{\mathcal{M}}_{g,n+1}} \operatorname{TF}_{g,n+1}^\chi \arrow[rr, "\operatorname{St}"] \arrow[rd, "\psi_1"']& & \Phi^*\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n+1}} \operatorname{TF}_{g,n+1}^\chi=:\mathcal{C}_2 \arrow[ld, "\psi_2"] \\ & \operatorname{TF}_{g,n+1}^\chi & \end{tikzcd}\tag{53}\] Fix a \(\psi_2\)-relative ample line bundle \(\mathcal{A}\) on \(\mathcal{C}_2\) and set \(\widehat{\mathcal{A}}:=\operatorname{St}^*(\mathcal{A})\). On \(\mathcal{C}_1\) there is a universal sheaf \(\mathcal{I}_{g,n+1}\) whose restriction on a geometric fiber \(\psi_1^{-1}(C,I)=C\) is equal to \(I\).

Let us first show that \(\mathcal{U}_{\geq 0}\) is open. Pick a geometric point \((C,I)\) of \(\operatorname{TF}_{g,n+1}^\chi\) that belongs to \(\mathcal{U}_{\geq 0}(k)\). We will show that \(\mathcal{U}_{\geq 0}\) contains an open neighborhood of \((C,I)\).

If \(C\) does not have any exceptional component, i.e. if \(\operatorname{st}_C:C\to C^{\operatorname{st}}\) is an isomorphism, then the same will be true on an open neighborhood of \(C\), which then will belong to \(\mathcal{U}_{\geq 0}\).

Assume now that \(C\) has an exceptional component \(E\), and consider the exact sequence (see 3 ) \[\label{E:I-resE} 0\to \leftindex_{E^c}I \to I \to I_{E}\to 0.\tag{54}\] Since the stabilization morphism \(\operatorname{st}_C:C\to C^{\operatorname{st}}\) is finite on \(E^c\), the line bundle \(\widehat{\mathcal{A}}_C:=\widehat{\mathcal{A}}_{|\psi_1^{-1}(C,I)=C}\) is ample on \(E^c\). Hence, there exists a constant \(N_C\gg 0\) such that \[\label{E:van-H1} H^1(C,\leftindex_{E^c}I\otimes \widehat{\mathcal{A}}_C^{m})=0 \text{ for any } m\geq N_C.\tag{55}\] By replacing \(\operatorname{TF}_{g,n+1}^\chi\) with a quasi-compact open neighboorhood of \((C,I)\), we may assume that \(N_C:=N\) is independent of \(C\). By taking the long exact sequence in cohomology associated to 54 and using 55 , we deduce that \[\label{E:equa-H1} H^1(C, I\otimes \widehat{\mathcal{A}}_C^{m})\cong H^1(C, I_E\otimes \widehat{\mathcal{A}}_C^{m})=H^1(E, I_E) \text{ for any } m\geq N,\tag{56}\] where we have used that \(\widehat{\mathcal{A}}_C\) is trivial on \(E\) since \(E\) is contracted by \(\operatorname{st}_C\). Therefore, we deduce that \[\label{E:U-vanH1} (C,I)\in \mathcal{U}_{\geq 0}(k) \Longleftrightarrow \chi(I_E)\geq 0 \Longleftrightarrow H^1(C, I\otimes \widehat{\mathcal{A}}_C^{m})=H^1((\mathcal{I}_{g,n+1}\otimes \widehat{\mathcal{A}}^m)_{|\psi_1^{-1}(C,I)})=0 \text{ for any } m\geq N.\tag{57}\] By the semicontinuity of cohomology, the last condition of 57 is open and therefore it is satisfied in an open neighboorhood of \((C,I)\).

The proofs for the other three loci are similar, with the following replacements:

\(\bullet\) For \(\mathcal{U}_{>0}\), one replaces the universal sheaf \(\mathcal{I}_{g,n+1}\) with the second reflexive hull \(\mathcal{I}_{g,n+1}^{[2]}\) of \(\mathcal{I}_{g,n+1}\), whose restriction to a geometric fiber \(\psi_1^{-1}(C,I)\) is equal to \(I^{[2]}\). Using that \((I^{[2]})_E=(I_E)^{2}\), Condition 57 becomes \[\label{E:U62-vanH1} (C,I)\in \mathcal{U}_{> 0}(k) \Longleftrightarrow \chi(I_E)> 0 \Longleftrightarrow H^1(C, I^{[2]}\otimes \widehat{\mathcal{A}}_C^{m})=H^1((\mathcal{I}_{g,n+1}^{[2]}\otimes \widehat{\mathcal{A}}^m)_{|\psi_1^{-1}(C,I)})=0 \text{ for any } m\geq N.\tag{58}\]

\(\bullet\) For \(\mathcal{V}_{>0}\), one replaces the universal sheaf \(\mathcal{I}_{g,n+1}\) with its dual sheaf \(\mathcal{I}_{g,n+1}^*={\mathcal{H}om}(\mathcal{I}_{g,n+1},\omega_{\psi_1})\), whose restriction to a geometric fiber \(\psi_1^{-1}(C,I)\) is equal to \(I^*={\mathcal{H}om}(I,\omega_C)\). Using that \((I^*)_E=(\leftindex_{E}I)^{*}={\mathcal{H}om}(\leftindex_{E}I,\omega_E)\) by Lemma 2, Condition 57 becomes \[\label{E:V-vanH1} \begin{align} & (C,I)\in \mathcal{V}_{\geq \chi}(k) \Longleftrightarrow \chi(I_{E^c})\geq \chi \Longleftrightarrow 0\geq \chi(\leftindex_{E}I)=-\chi((\leftindex_{E}I)^*)=-\chi((I^*)_E) \Longleftrightarrow \\ & \Longleftrightarrow H^1(C, I^{*}\otimes \widehat{\mathcal{A}}_C^{m})=H^1((\mathcal{I}_{g,n+1}^{*}\otimes \widehat{\mathcal{A}}^m)_{|\psi_1^{-1}(C,I)})=0 \text{ for any } m\geq N. \end{align}\tag{59}\]

\(\bullet\) For \(\mathcal{V}_{>0}\), one replaces the universal sheaf \(\mathcal{I}_{g,n+1}\) with the second reflexive hull \((\mathcal{I}_{g,n+1}^*)^{[2]}\) of the dual sheaf \(\mathcal{I}_{g,n+1}^*\), whose restriction to a geometric fiber \(\psi_1^{-1}(C,I)\) is equal to \((I^*)^{[2]}\). Using that \(((I^*))^{[2]}_E=((I^*)_E)^2=((\leftindex_{E}I)^{*})^2\) by Lemma 2, Condition 57 becomes \[\label{E:V62-vanH1} \begin{align} & (C,I)\in \mathcal{V}_{> \chi}(k) \Longleftrightarrow \chi(I_{E^c})> \chi \Longleftrightarrow 0> \chi(\leftindex_{E}I)=-\chi((\leftindex_{E}I)^*)=-\chi((I^*)_E) \Longleftrightarrow \\ & \Longleftrightarrow H^1(C, (I^{*})^{[2]}\otimes \widehat{\mathcal{A}}_C^{m})=H^1(((\mathcal{I}_{g,n+1}^{*})^{[2]}\otimes \widehat{\mathcal{A}}^m)_{|\psi_1^{-1}(C,I)})=0 \text{ for any } m\geq N. \end{align}\tag{60}\]

Part [T:univ-TF2]: let \((C,I)\) be a geometric point of \(^{o}\operatorname{TF}_{g,n+1}^\chi\) and consider the stabilization \(\operatorname{st}:C\to C^{\operatorname{st}}\). Lemma 8 implies that \(\operatorname{st}_*(I)\in \operatorname{TF}_{C^{\operatorname{st}}}^\chi\).

In order to define the morphism \(\widehat{\Phi}\), consider the restriction of the diagram in 53 to the open substack \(^{o}\operatorname{TF}_{g,n+1}^\chi\): \[\label{E:univ-stabo} \begin{tikzcd} \mathcal{C}_1^o:=\overline{\mathcal{C}}_{g,n+1}\times_{\overline{\mathcal{M}}_{g,n+1}} {^{o}\operatorname{TF}_{g,n+1}^\chi} \arrow[rr, "\operatorname{St}^o"] \arrow[rd, "\psi_1^o"']& & \Phi^*\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n+1}} {^{o}\operatorname{TF}_{g,n+1}^\chi}=:\mathcal{C}_2^o \arrow[ld, "\psi^o_2"] \\ & ^{o}\operatorname{TF}_{g,n+1}^\chi & \end{tikzcd}\tag{61}\] On \(\mathcal{C}_1^o\) there is a universal sheaf \(\mathcal{I}_{g,n+1}\) whose restriction to a geometric fiber \(\psi_1^{-1}(C,I)=C\) is equal to \(I\). The existence of the morphism \(\widehat{\Phi}\) lying over \(\Phi\) and given on geometric points by \(\widehat\Phi(C,I)=(C^{\operatorname{st}}, \operatorname{st}_*(I))\) will follow from the following

Claim: The sheaf \(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})\) is a relative rank-one torsion-free sheaf on \(\mathcal{C}_2^o/^{o}\operatorname{TF}_{g,n+1}^\chi\) such that \(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})_{|(\psi_2^o)^{-1}(C,I)}=\operatorname{st}_*(I)\).

Indeed, fix a \(\psi_2\)-relative ample line bundle \(\mathcal{A}\) on \(\mathcal{C}_2^o\) and set \(\widehat{\mathcal{A}}:=\operatorname{St}^*(\mathcal{A})\). The flatness of \(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})\) over \(^{o}\operatorname{TF}_{g,n+1}^\chi\) is equivalent to showing that, locally on \(^{o}\operatorname{TF}_{g,n+1}^\chi\), the sheaf \[\label{E:sheaf-A} (\psi_2^o)_*(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})\otimes \mathcal{A}^m)=(\psi_1^o)_*(\mathcal{I}_{g,n+1}\otimes \widehat{\mathcal{A}}^m)\tag{62}\] is locally free for any \(m\gg 0\). This follows from the vanishing on the right hand side of 57 , using that \(^{o}\operatorname{TF}_{g,n+1}^\chi\subseteq \mathcal{U}_{\geq 0}\). Moreover, the vanishing on the right hand side of 57 implies also that the sheaf \((\psi_1^o)_*(\mathcal{I}_{g,n+1}\otimes \widehat{\mathcal{A}}^m)\) commutes with base change (locally on \(^{o}\operatorname{TF}_{g,n+1}^\chi\)) for any \(m\gg 0\), which in turn implies that the sheaf \(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})\) satisfies base change. In particular, the fiber of \(\operatorname{St}^o_*(\mathcal{I}_{g,n+1})\) over a geometric point \((C,I)\) of \(^{o}\operatorname{TF}_{g,n+1}^\chi\) is equal to \(\operatorname{st}_*(I)\), which is a rank-one torsion-free sheaf on \(C^{\operatorname{st}}\) of Euler characteristic \(\chi\) by Lemma 8[L:push-st1]. This concludes the proof of Claim.

The morphism \(\widehat\Upsilon\) is defined as the fiber product of \(\widehat{\Phi}\) and of the morphism \(^{o}\operatorname{TF}_{g,n+1}^\chi\to \overline{\mathcal{M}}_{g,n+1}\xrightarrow{\Upsilon} \overline{\mathcal{C}}_{g,n}\), and therefore it lies over \(\Upsilon\) and it is given on geometric points by \(\widehat{\Upsilon}(C,I)=(C^{\operatorname{st}},\operatorname{st}(p_{n+1}),\operatorname{st}_*(I))\).

In order to define the section \(\widehat{\sigma_i}'\) of \(\widehat{\Phi}\), consider the following diagram induced by the section \(\sigma_i'\) of \(\Phi\): \[\label{E:univ-si} \begin{tikzcd} \mathcal{D}_1:=(\sigma_i')^*\overline{\mathcal{C}}_{g,n+1}\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}^\chi \arrow[rr, "\overline{\operatorname{St}}"] \arrow[rd, "\phi_1"']& & \overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}^\chi=:\mathcal{D}_2 \arrow[ld, "\phi_2"] \\ & \operatorname{TF}_{g,n}^\chi .& \end{tikzcd}\tag{63}\] The fiber of \(\overline{\operatorname{St}}\) over a geometric point \((C,I)\in \operatorname{TF}_{g,n}^\chi\) is equal to the stabilization morphism \(\operatorname{st}=\operatorname{st}_{B_{p_i}(C)}:B_{p_i}(C)=\phi_1^{-1}(C,I)\to \phi_2^{-1}(C,I)=C\) of the bubbling of \(C\) at the \(i\)-th marked point \(p_i\).

Consider now the sheaf \(\overline{\operatorname{St}}^*(\mathcal{I}_{g,n})\) on \(\mathcal{D}_1\), where \(\mathcal{I}_{g,n}\) is the universal sheaf on \(\mathcal{D}_2\). Since the sheaf \(\mathcal{I}_{g,n}\) is locally free around the image of the section \(\sigma_i\), it follows that \(\overline{\operatorname{St}}^*(\mathcal{I}_{g,n})\) is a relative rank-one torsion-free sheaf on \(\mathcal{D}_1/\operatorname{TF}_{g,n}^\chi\), whose fiber over a geometric point \((C,I)\) of \(\operatorname{TF}_{g,n}^{\chi}\) is equal to \(\operatorname{st}^*(I)\). Therefore, the sheaf \(\overline{\operatorname{St}}^*(\mathcal{I}_{g,n})\) defines a morphism \(\widehat{\sigma_i}': \operatorname{TF}_{g,n}^\chi\to \operatorname{TF}_{g,n+1}^\chi\) lying over \(\sigma_i'\) and which is given on geometric points by \(\widehat{\sigma_i}'(C,I)=(B_{p_i}(C),\operatorname{st}^*(I))\). Moreover, if we denote by \(E\) the exceptional component of \(B_{p_i}(C)\) (so that \(\operatorname{st}\) identifies \(E^c\) with \(C\)) then we have that \[{ \left\{\begin{align} & \operatorname{st}^*(I)_{E}=\mathcal{O}_E\Rightarrow \chi(\operatorname{st}^*(I)_E)=1,\\ & \operatorname{st}^*(I)_{E^c}=I\Rightarrow \chi(\operatorname{st}^*(I)_{E^c})=\chi, \end{align}\right.}\] which implies that \(\widehat{\sigma_i}'\) factors through the open subset \(^{o}\operatorname{TF}_{g,n+1}^\chi\). Finally, the fact that \(\widehat{\sigma_i}'\) is a section of \(\widehat{\Phi}\) follows from the fact that \(\overline{\operatorname{St}}_*(\overline{\operatorname{St}}^*(\mathcal{I}_{g,n}))=\mathcal{I}_{g,n}\).

Part [T:univ-TF2bis]: the fact that \(\widehat{\Upsilon}\) is a good moduli space morphism follows from Proposition 24, using that this property can be checked fpqc locally on the codomain by [42]. The last properties of \(\widehat{\Upsilon}\) and \(\widehat{\Phi}\) follows from the fact that a good moduli space morphism is universally closed, \(\Theta\)-complete and S-complete (see [44]) and that \(\widehat{\Phi}=\widehat{\pi}\circ \widehat{\Upsilon}\) with \(\widehat{\pi}\) being proper and representable.

Part [T:univ-TF3]: this follows from Proposition 24, using that the properties in question can be checked fpqc locally on the codomain.

Part [T:univ-TF4]: we will prove the statement for \(^{+}\operatorname{TF}_{g,n+1}^\chi\), and we will leave to the reader the task of figuring out the necessary (small) adjustments needed in order to prove the statement for \(^{-}\operatorname{TF}_{g,n+1}^\chi\). Set \(\widehat{\Upsilon}^+:=\widehat{\Upsilon}_{|^{+}\operatorname{TF}_{g,n+1}^\chi}\) and \(\widehat{\Phi}^+:=\widehat{\Phi}_{|^{+}\operatorname{TF}_{g,n+1}^\chi}\) Using the universal property of \(\mathbb{P}(\mathcal{I}_{g,n})\) (see [43]), it is enough to check the following properties:

(a) \(\widehat{\Phi}^+\) is a family of quasi-stable \(n\)-pointed curves and \(\widehat{\Upsilon}^+\) is its stabilization morphism;

(b) the line bundle \(\widehat{\sigma}_{n+1}^*(\mathcal{I}_{g,n+1})\) has degree \(1\) on every exceptional component of \(\widehat{\Upsilon}^+\);

(c) \((\widehat{\Upsilon}^+)_*(\widehat{\sigma}_{n+1}^*(\mathcal{I}_{g,n+1}))=\mathcal{I}_{g,n}\).

Observe that the morphism \(\widehat{\Phi}^+\) is proper and representable, being the composition of \(\widehat{\Upsilon}^+\) (which is proper and representable by Part [T:univ-TF3]) with the proper and representable morphism \(\widehat{\pi}\). Proposition 24 implies that the geometric fiber of \(\widehat{\Phi}^+\) over \((D,J)\in \operatorname{TF}_{g,n}^\chi(k=\overline{k})\) is a quasi-stable \(n\)-pointed curve which is obtained by bubbling \(D\) into the nodes on which \(J\) is not free and, furthermore, that \(\widetilde{\Upsilon}^+\) is, on geometric fibers, the stabilization morphism. And finally, since the domain and codomain of \(\widehat{\Phi}^+\) are regular (see [19]), by miracle flatness we have that \(\widehat{\Phi}^+\) is flat because it is equidimensional of relative dimension one. This shows that \(\widehat{\Phi}^+\) is a family of quasi-stable \(n\)-pointed curves and part (a) is established.

Consider now an exceptional component of \(\widehat{\Upsilon}^+\), which, as noted above, is the preimage \((\widehat{\Upsilon}^+)^{-1}(D,p,J)\), for \((D,p,J)\in \overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}^\chi\) such that \(p\) is a node of \(D\) and \(J\in \operatorname{TF}_D^\chi\) is not free at \(p\). Denote by \(B_p(D)\) the bubbling of \(D\) at \(p\) and by \(\operatorname{st}:B_p(D)\to D\) the stabilization morphism which contracts the exceptional component \(E_p\cong \mathbb{P}^1\) of the bubbling into \(p\). From the proof of Proposition 24[P:push-loc4] it follows that the exceptional component \((\widehat{\Upsilon}^+)^{-1}(D,p,J)\) is the rational curve \(E_p=\mathbb{P}^1\hookrightarrow \operatorname{TF}_{B_p(D)}^\chi\) corresponding to the relative rank-one torsion-free sheaf \(\mathcal{I}^+:=\pi_1^*(I^+)\otimes \mathcal{O}(\Delta)\) of Euler characteristic \(\chi\) on \(B_p(D)\times E_p\to E_p\), where \(\pi_1\) is the projection onto the first factor, \(I^+\in \operatorname{TF}_{B_p(D)}^{\chi-1}\) is free at \(E_p\cap E_p^c\) with \((\operatorname{st}_{|E_p^c})_*(I^+_{E_p^c})=J\) and \(\Delta\subset E_p\times E_p\subset B_p(D)\times E_p\) is the diagonal. Therefore, the restriction of the line bundle \(\widehat{\sigma}_{n+1}^*(\mathcal{I}_{g,n+1})\) on \((\widehat{\Upsilon}^+)^{-1}(D,p,J)\cong E_p=\mathbb{P}^1\) is equal to \(p_{n+1}^*(\mathcal{I}^+)=p_{n+1}^*(\pi_1^*(I^+)\otimes \mathcal{O}(\Delta))=p_{n+1}^*\mathcal{O}(\Delta)=\mathcal{O}_{\mathbb{P}^1}(p_{n+1})\) using that \(p_{n+1}\in E_p\) and \(\pi_1^*(I^+)\) is trivial on \(E_p\). Part (b) follows from this.

It remains to prove Part (c). Let \((\operatorname{TF}_{g,n}^\chi)^o\) be the open substack of \(\operatorname{TF}_{g,n}^\chi\) whose geometric points are \((C,L)\) where \(C\in \overline{\mathcal{M}}_{g,n}(k)\) and \(L\) is a line bundle on \(C\). Note that \((\operatorname{TF}_{g,n}^\chi)^o\) is a big open subset of \(\operatorname{TF}_{g,n}^\chi\), i.e. its complement has codimension at least two. By the above description of \(\widehat{\Upsilon}^+\), it follows that \(\widehat{\Upsilon}^+\) is an isomorphism over \(\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} (\operatorname{TF}_{g,n}^\chi)^o\). Moreover, the composition \(\widehat{\sigma}_{n+1}\circ (\widehat{\Upsilon}^+)^{-1}\) is given on geometric points by \[\begin{align} \widehat{\sigma}_{n+1}\circ (\widehat{\Upsilon}^+)^{-1}: \overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} (\operatorname{TF}_{g,n}^\chi)^o& \to \overline{\mathcal{C}}_{g,n+1}\times_{\overline{\mathcal{M}}_{g,n+1}} ^{o}\operatorname{TF}_{g,n+1}^\chi\\ (C,p,L) & \mapsto (\Upsilon^{-1}(C,p),p_{n+1}, \operatorname{st}^*(L)) \end{align}\] where \(\Upsilon^{-1}\) is the inverse of the isomorphism \(\Upsilon\) of 51 and \(\operatorname{st}:\Upsilon^{-1}(C,p)\to C\) is the map that forgets the last marked point and stabilizes. From the explicit form of this map, it is clear that \((\widehat{\sigma}_{n+1}\circ (\widehat{\Upsilon}^+)^{-1})^*(\mathcal{I}_{g,n+1})=\mathcal{I}_{g,n}\). This is equivalent to saying that \[\widehat{\Upsilon}^+_*(\widehat{\sigma}_{n+1}^*(\mathcal{I}_{g,n+1}))_{|\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} (\operatorname{TF}_{g,n}^\chi)^o}=(\mathcal{I}_{g,n})_{|\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} (\operatorname{TF}_{g,n}^\chi)^o}.\] We now conclude that the two sheaves agree everywhere by [3] using that: \(\widehat{\Upsilon}^+_*(\widehat{\sigma}_{n+1}^*(\mathcal{I}_{g,n+1}))\) is a family of rank-one torsion-free sheaves on \(\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}}\operatorname{TF}_{g,n}^\chi\) by Parts (a) and (b) and by [43]; \((\operatorname{TF}_{g,n}^\chi)^o\subset \operatorname{TF}_{g,n}^\chi\) is a big open substack, \(\operatorname{TF}_{g,n}^\chi\) is regular (see [19]) and \(\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}}\operatorname{TF}_{g,n}^\chi\) is \(G_1\) and \(S_2\) (see [3]). ◻

Proof. Consider the two exact sequences (see 3 ) \[\label{E:2ex-seq} { \left\{\begin{align} & 0 \to \leftindex_{E^c} I \to I \to I_E\to 0, \\ & 0 \to \leftindex_{E} I \to I \to I_{E^c}\to 0. \\ \end{align}\right.}\tag{64}\] By applying \(\operatorname{st}_*\) to the first exact sequence of 64 and using that \(R^1\operatorname{st}_*(\leftindex_{E^c} I)=0\) since \(\operatorname{st}_{|E^c}:E^c\to C^{\operatorname{st}}\) is a partial normalization (and hence a finite map), we deduce that \[\label{E:equa1} R^1\operatorname{st}_*(I)\cong R^1\operatorname{st}_*(I_E)=\mathcal{O}_q^{\oplus h^1(E,I_E)}=0 \text{ since } \chi(I_E)\geq 0.\tag{65}\] By applying \(\operatorname{st}_*\) to the second exact sequence of 64 and using 65 , we get the exact sequence \[\label{E:equa2} 0\to \operatorname{st}_*(\leftindex_{E}I)=\mathcal{O}_q^{\oplus h^0(E,\leftindex_{E}I)}\to \operatorname{st}_*(I)\to \operatorname{st}_*(I_{E^c})\to R^1\operatorname{st}_*(\leftindex_{E}I)=\mathcal{O}_q^{\oplus h^1(E,\leftindex_{E}I)}\to 0.\tag{66}\] Since \(\operatorname{st}_*(I_{E^c})\) is a rank-one torsion-free sheaf on \(C^{\operatorname{st}}\) with \(\chi(\operatorname{st}_*(I_{E^c}))=\chi(I_{E^c})\) (because \(\operatorname{st}_{|E^c}:E^c\to C\) is a partial normalization map), we deduce from 66 that \(\operatorname{st}_*(I)\) is a rank-one sheaf on \(C^{\operatorname{st}}\) of Euler characteristic \[\chi(\operatorname{st}_*(I))=\chi(I_{E^c})+\chi(\leftindex_{E} I)=\chi(I).\] Moreover, \(\operatorname{st}_*(I)\) is torsion-free since \[\label{E:equa3} \mathcal{O}_q^{\oplus h^0(E,\leftindex_{E}I)}=0 \text{ because } \chi(\leftindex_{E}I)=\chi(I)-\chi(I_{E^c})\leq 0.\tag{67}\] Finally, the second exact sequence in Part [L:push-st3] follows from 66 , while the first exact sequence follows by applying \(\operatorname{st}_*\) to the first sequence of 64 and using that \(R^1\operatorname{st}_*(\leftindex_{E^c}I)=0\) since \(\operatorname{st}_{|E^c}\) is a finite map. ◻

Proof. Part [P:push-loc1] is obvious.

Part [P:push-loc2]: let us compute the fiber \(\widehat{\Upsilon}^{-1}(D,p,J)\). First of all, \(\Upsilon^{-1}(D,p)\) is the bubbling \(B_p(D)\) of \(D\) at \(p\), i.e. the stable \(n+1\)-pointed curve obtained by inserting a \(E\cong \mathbb{P}^1\) at \(p\) and putting the marked point \(p_{n+1}\) on the exceptional component \(E\). Therefore, \[\widehat{\Upsilon}^{-1}(D,p,J)=\{(B_p(D),I)\: : I\in ^{o}\operatorname{TF}^\chi_{B_p(D)} \text{ with } \operatorname{st}_*(I)=J\}.\] Take now \(I\in ^{o}\operatorname{TF}^\chi_{B_p(D)}\) such that \(\operatorname{st}_*(I)=J\). Since \(J\) is free at \(p\), we must have by Lemma 8[L:push-st3] that \[h^0(E,I_E)>0 \text{ and } h^1(E,\leftindex_{E}I)>0,\] which is only possible if \(I\) is free at \(E\cap E^c\) and \(I_E=\mathcal{O}_E\). This implies that \(I\) is the pull-back via \(\operatorname{st}\) of a sheaf on \((B_p(D))^{\operatorname{st}}=D\), and then, by the projection formula, we must have that \(I=\operatorname{st}^*(J)\). Hence, we get that \[\widehat{\Upsilon}^{-1}(D,p,J)=\{(B_p(D),\operatorname{st}^*(J))\}.\] This also shows that \(\widehat{\Upsilon}\) is an isomorphism locally above \((D,p,J)\), since a local inverse is given by the morphism \(\overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}\to ^{o}\operatorname{TF}_{g,n+1}^\chi\) induced by the pull-back of the universal sheaf \(\mathcal{I}_{g,n}\) on the family \[(\Upsilon^{-1})^*(\overline{\mathcal{C}}_{g,n})\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}^\chi\to \overline{\mathcal{C}}_{g,n}\times_{\overline{\mathcal{M}}_{g,n}} \operatorname{TF}_{g,n}.\]

Part [P:push-loc3]: observe that \(\Upsilon^{-1}(D,p)\) is the bubbling \(C:=B_p(D)\) of \(D\) at the point \(p\), which is a marked point of \(D\). Hence \(C=E\cup E^c\), where \(E\cong \mathbb{P}^1\) is is the exceptional component, \(E^c\) is canonically isomorphic to \(D\) via the restriction of the stabilization morphism \(\operatorname{st}:C\to D\) and \(E\) and \(E^c\) meet in a unique point \(q\) which is a node of \(C\). The stack \(^{o}\operatorname{TF}_C^\chi\) admits the following stratification \[\label{E:TF-strat1} ^{o}\operatorname{TF}_C^\chi=^{+}\operatorname{TF}_C^\chi \sqcup ^{-}\operatorname{TF}_C^\chi \sqcup (^{o}\operatorname{TF}_C^\chi)^{cl},\tag{68}\] where:

\(\bullet\) \(^{+}\operatorname{TF}_C^\chi\) and \(^{-}\operatorname{TF}_C^\chi\) are the open substacks given by \[\begin{align} & ^{+}\operatorname{TF}_C^\chi:=\{I\in \operatorname{TF}_C^\chi\: : \chi(I_E)\geq 1 \text{ and } \chi(I_{E^c})\geq \chi\}, \\ & ^{-}\operatorname{TF}_C^\chi:=\{I\in \operatorname{TF}_C^\chi\: : \chi(I_E)\geq 0 \text{ and } \chi(I_{E^c})\geq \chi+1\}; \end{align}\]

\(\bullet\) \((^{o}\operatorname{TF}_C^\chi)^{cl}\) is the closed substack parametrizing sheaves that are not free at \(q\) and hence it is given by \[(^{o}\operatorname{TF}_C^\chi)^{cl}:=\{K\oplus \mathcal{O}_E(-1)\: : K\in \operatorname{TF}_D^\chi\}.\] Using Lemma 8, the map \(\operatorname{st}_*:^{o}\operatorname{TF}_C^\chi\to \operatorname{TF}_D^\chi\) is given on the above three strata by \[\label{E:st-strat1} \begin{align} \operatorname{st}_*:^{+}\operatorname{TF}_C^\chi& \longrightarrow \operatorname{TF}_D^\chi \\ I &\mapsto \operatorname{st}_*(I_{E^c}) \end{align} \quad \text{ and } \quad \begin{aligned} \operatorname{st}_*:^{-}\operatorname{TF}_C^\chi& \longrightarrow \operatorname{TF}_D^\chi\\ I &\mapsto \operatorname{st}_*(\leftindex_{E^c}I) \end{aligned} \quad \text{ and } \quad \begin{align} \operatorname{st}_*:(^{o}\operatorname{TF}_C^\chi)^{cl}& \longrightarrow \operatorname{TF}_D^\chi\\ K\oplus \mathcal{O}_E(-1) &\mapsto K. \end{align}\tag{69}\] By putting together 68 and 69 , we deduce that \[\label{E:inv-Ups1} \widehat{\Upsilon}^{-1}(D,p,J)=\{(C,I)\: : J\oplus \mathcal{O}_E(-1)\in \overline{\{I\}}\}.\tag{70}\]

Hence, the map \(\widehat{\Upsilon}\), étale locally on \((D,p,J)\) and up to smooth factors and \(\operatorname{\mathbb{G}_m}\)-rigidification, is modeled on the map of deformation functors (by [45]) \[\label{E:Def-Ups1} \frac{\operatorname{Def}_q(C,J\oplus \mathcal{O}_E(-1))}{\operatorname{\mathbb{G}_m}}\xrightarrow{\widehat{\Upsilon}_*} \operatorname{Def}_p(D,p,J),\tag{71}\] where \(\operatorname{Def}_q(C,J\oplus \mathcal{O}_E(-1))\) is the local deformation functor of the pair \((C,J\oplus \mathcal{O}_E(-1))\) at \(q\), the action of \(\operatorname{\mathbb{G}_m}\) is induced by the scalar multiplication of \(\operatorname{\mathbb{G}_m}\) on \(E\), and \(\operatorname{Def}_p(D,p,J)\) is the local deformation functor of the triple \((D,p,J)\) at \(p\). Moreover, since the morphism \(\widehat{\Upsilon}\) sits above the morphism \(\Upsilon\), the map in 71 fits into the following commutative diagram of local deformation functors \[\label{E:Def-diag1} \begin{tikzcd} \displaystyle \frac{\operatorname{Def}_q(C,J\oplus \mathcal{O}_E(-1))}{\operatorname{\mathbb{G}_m}} \arrow["\widehat{\Upsilon}_*", r] \arrow[d] & \operatorname{Def}_p(D,p,J) \arrow[d, "\cong"] \\ \operatorname{Def}_q(C) \arrow["\Upsilon_*", "\cong" swap, r] & \operatorname{Def}_p(D,p) \end{tikzcd}\tag{72}\] where the vertical maps are forgetful morphism, the bottom horizontal map is an isomorphism since \(\Upsilon\) is an isomorphism and the vertical right map is an isomorphism since \(J\) is free at \(p\) (because \(p\) is a smooth point of \(D\)) and hence its local deformations are trivial. Using the above diagram and the description of the local deformations of a pair consisting of a nodal curve and a rank-one torsion-free sheaf in [46], we deduce that the miniversal deformation spaces for the map in 71 are the completion at the origin of the following map \[\label{E:model-Ups1} \left[\frac{\operatorname{Spec}k[w,\overline{w}]}{\operatorname{\mathbb{G}_m}}\right]\to \operatorname{Spec}k[t],\tag{73}\] where \(\operatorname{\mathbb{G}_m}\) acts via \(\lambda\cdot (w,\overline{w}):=(\lambda w, \lambda^{-1}\overline{w})\) and the map is given by identifying \(t\) with \(w\overline{w}\). If we set \(V_1:=\langle w\rangle\), then the map in 73 is exactly the map \(\mu\) in the statement of Part [P:push-loc3]. Moreover, it follows again from loc. cit. that the local model for morphism \(\widehat{\Upsilon}^+\) (resp. \(\widehat{\Upsilon}^-\)) is the restriction of 73 to the open substack \(\{w\neq 0\}\) (resp. \(\{\overline{w}\neq 0\}\)), which is exactly the map \(\mu^+\) (resp. \(\mu^-\)) in the statement of Part [P:push-loc3]. The proof of Part [P:push-loc3] is now complete.

Part [P:push-loc4]: observe that \(\Upsilon^{-1}(D,p)\) is the bubbling \(C:=B_p(D)\) of \(D\) at the node \(p\) of \(D\). Hence \(C=E\cup E^c\), where \(E\cong \mathbb{P}^1\) is is the exceptional component, \(E^c\) is canonically isomorphic to the partial normalization of \(D\) at \(p\) via the restriction of the stabilization morphism \(\operatorname{st}:C\to D\) and \(E\) and \(E^c\) meet in two points \(\{q_1,q_2\}\) which are nodes of \(C\). The stack \(^{o}\operatorname{TF}_C^\chi\) admits the following stratification \[\label{E:TF-strat2} ^{o}\operatorname{TF}_C^\chi=(^{o}\operatorname{TF}_C^\chi)^{op}\sqcup (^{+}\operatorname{TF}_C^\chi)^{cl} \sqcup (^{-}\operatorname{TF}_C^\chi)^{cl} \sqcup (^{o}\operatorname{TF}_C^\chi)^{cl},\tag{74}\] where:

\(\bullet\) \((^{o}\operatorname{TF}_C^\chi)^{op}\) is the open substack consisting of sheaves that are free at \(q_1\) and \(q_2\) and they are trivial on \(E\), or equivalently \[(^{o}\operatorname{TF}_C^\chi)^{op}=^{+}\operatorname{TF}_C^\chi\cap ^{-}\operatorname{TF}_C^\chi=\{I\in \operatorname{TF}_{C}^\chi : \chi(I_{E})= 1 \text{ and } \chi(I_{E^c})= \chi+1\};\]

\(\bullet\) \((^{+}\operatorname{TF}_C^\chi)^{cl}\) and \((^{-}\operatorname{TF}_C^\chi)^{cl}\) are the locally closed substacks given by \[{ \left\{\begin{align} & (^{+}\operatorname{TF}_C^\chi)^{cl}:=^{+}\operatorname{TF}_C^\chi- (^{o}\operatorname{TF}_C^\chi)^{op}=\{I\in \operatorname{TF}_{C}^\chi : \chi(I_{E})\geq 1 \text{ and } \chi(I_{E^c})= \chi\}, \\ & (^{-}\operatorname{TF}_{C}^\chi)^{cl}:=^{-}\operatorname{TF}_C^\chi- (^{o}\operatorname{TF}_C^\chi)^{op}=\{I\in \operatorname{TF}_{C}^\chi : \chi(I_{E})=0 \text{ and } \chi(I_{E^c})\geq \chi+1\}; \\ \end{align}\right.}\]

\(\bullet\) \((^{o}\operatorname{TF}_C^\chi)^{cl}\) is the closed substack parametrizing sheaves that are not free at \(\{q_1,q_2\}\) and hence it is given by \[(^{o}\operatorname{TF}_C^\chi)^{cl}:=\{K\oplus \mathcal{O}_E(-1)\: : K\in \operatorname{TF}_{E^c}^\chi\}.\] In order to describe the morphisms \(\operatorname{st}_*\), we partition the codomain into an open and closed substack as follows \[\operatorname{TF}_{D}^\chi=(\operatorname{TF}_D^\chi)^{op}\sqcup (\operatorname{TF}_D^\chi)^{cl},\] given by, respectively \[{ \left\{\begin{align} &(\operatorname{TF}_D^\chi)^{op}:=\{J\in \operatorname{TF}_D^\chi : J \text{ is free at } p\}, \\ &(\operatorname{TF}_D^\chi)^{cl}:=\{J\in \operatorname{TF}_D^\chi : J \text{ is not free at } p\}. \end{align}\right.}\] Since \((^{o}\operatorname{TF}_C^\chi)^{op}\) consists of sheaves that are free at \(q_1\) and \(q_2\) and are trivial on \(E\), the push-forward \(\operatorname{st}_*\) induces an isomorphism \[\label{E:st-open} \operatorname{st}_*: (\operatorname{TF}_C^\chi)^{op} \xrightarrow{\cong} (\operatorname{TF}_D^\chi)^{op},\tag{75}\] with inverse given by the pull-back map \(\operatorname{st}^*\). Using Lemma 8, the map \(\operatorname{st}_*\) on the other three strata is given by \[\label{E:st-strat2} \begin{align} \operatorname{st}_*:(^{+}\operatorname{TF}_C^\chi)^{cl}& \longrightarrow (\operatorname{TF}_D^\chi)^{cl} \\ I &\mapsto \operatorname{st}_*(I_{E^c}) \end{align} \quad \text{ and } \quad \begin{aligned} \operatorname{st}_*:(^{-}\operatorname{TF}_C^\chi)^{cl}& \longrightarrow (\operatorname{TF}_D^\chi)^{cl}\\ I &\mapsto \operatorname{st}_*(\leftindex_{E^c}I) \end{aligned} \quad \text{ and } \quad \begin{align} \operatorname{st}_*:(^{o}\operatorname{TF}_C^\chi)^{cl}& \longrightarrow (\operatorname{TF}_D^\chi)^{cl}\\ K\oplus \mathcal{O}_E(-1) &\mapsto \operatorname{st}_*(K). \end{align}\tag{76}\] Now observe that, since \(\operatorname{st}_{|E^c}:E^c\to D\) is the partial normalization of \(D\) at the node \(p\), we have an isomorphism \[\label{E:iso-clst} (\operatorname{st}_{|E^c})_*:\operatorname{TF}^\chi_{E^c} \xrightarrow{\cong} (\operatorname{TF}_D^\chi)^{cl}.\tag{77}\] Hence, since \(J\in (\operatorname{TF}_D^\chi)^{cl}\) by assumption, there exists a unique \(\widetilde{J}\in \operatorname{TF}_{E^c}^\chi\) such that \(\operatorname{st}_*(\widetilde{J})=J\).

By combining 75 and 76 , we deduce that \[\label{E:inv-Ups2} \widehat{\Upsilon}^{-1}(D,p,J)=\{(C,I)\: : \widetilde{J}\oplus \mathcal{O}_E(-1)\in \overline{\{I\}}\}.\tag{78}\]

Hence, the map \(\widehat{\Upsilon}\), étale locally on \((D,p,J)\) and up to smooth factors and \(\operatorname{\mathbb{G}_m}\)-rigidification, is modeled on the map of deformation functors (by [45]) \[\label{E:Def-Ups2} \frac{\operatorname{Def}_{\{q_1,q_2\}}(C,\widetilde{J}\oplus \mathcal{O}_E(-1))}{\operatorname{\mathbb{G}_m}}\xrightarrow{\widehat{\Upsilon}_*} \operatorname{Def}_p(D,p,J)=\operatorname{Def}_p(D,p)\times_{\operatorname{Def}_p(D)}\operatorname{Def}_p(D,J),\tag{79}\] where \(\operatorname{Def}_{\{q_1,q_2\}}(C,\widetilde{J}\oplus \mathcal{O}_E(-1))\) is the local deformation functor of the pair \((C,J\oplus \mathcal{O}_E(-1))\) at the two points \(\{q_1,q_2\}\), the action of \(\operatorname{\mathbb{G}_m}\) is induced by the scalar multiplication of \(\operatorname{\mathbb{G}_m}\) on \(E\), and \(\operatorname{Def}_p(D,p,J)\) is the local deformation functor of the triple \((D,p,J)\) at \(p\) which is the fiber product of the local deformation functor \(\operatorname{Def}_p(D,p)\) of the pair \((D,p)\) at \(p\) with the local deformation functor of \((D,J)\) at \(p\) over the local deformation functor of \(\operatorname{Def}_p(D)\) of \(D\) at \(p\). Moreover, since the morphism \(\widehat{\Upsilon}\) lies above the morphism \(\Upsilon\), the map in 79 fits into the following commutative diagram of local deformation functors \[\label{E:Def-diag2} \begin{tikzcd} \displaystyle \frac{\operatorname{Def}_{\{q_1,q_2\}}(C,\widetilde{J}\oplus \mathcal{O}_E(-1))}{\operatorname{\mathbb{G}_m}} \arrow["\widehat{\Upsilon}_*", r] \arrow[d] & \operatorname{Def}_p(D,p,J)=\operatorname{Def}_p(D,p)\times_{\operatorname{Def}_p(D)}\operatorname{Def}_p(D,J) \arrow[d] \\ \operatorname{Def}_{\{q_1,q_2\}}(C) \arrow["\Upsilon_*", "\cong" swap, r] & \operatorname{Def}_p(D,p) \end{tikzcd}\tag{80}\] where the vertical maps are forgetful morphism, the bottom horizontal map is an isomorphism since \(\Upsilon\) is an isomorphism. Using the above diagram and the description of the local deformations of a pair consisting of a nodal curve and a rank-one torsion-free sheaf in [46], we deduce that the miniversal deformation spaces for the diagram in 80 are the completions at the origin of the following diagram \[\label{E:model-Ups2} \begin{tikzcd} \displaystyle \left[\frac{\operatorname{Spec}k[w_1,\overline{w}_1, w_2, \overline{w}_2]}{\operatorname{\mathbb{G}_m}}\right] \arrow[rr] \arrow[dr]&& \displaystyle \operatorname{Spec}\frac{k[x,y,u,v]}{(xy-uv)} \arrow[ld]\\ & \operatorname{Spec}k[x,y] & \end{tikzcd}\tag{81}\] where \(\operatorname{\mathbb{G}_m}\) acts via \(\lambda\cdot (w_1,\overline{w}_1,w_2, \overline{w}_2):=(\lambda w_1, \lambda^{-1}\overline{w}_1, \lambda w_2, \lambda^{-1}\overline{w}_2)\), and the maps are given by the rules \[x=w_1\overline{w}_1, \;y=w_2\overline{w}_2, \;u=w_1\overline{w}_2,\;v=w_2\overline{w}_1.\] If we set \(V_2:=\langle w_1,w_2\rangle\), then the map in 81 is exactly the map \(\nu\) in the statement of Part [P:push-loc4]. Moreover, it follows again from loc. cit. that the local model for morphism \(\widehat{\Upsilon}^+\) (resp. \(\widehat{\Upsilon}^-\)) is the restriction of 73 to the open substack \(\{(w_1,w_2)\neq (0,0)\}\) (resp. \(\{(\overline{w}_1,\overline{w}_2)\neq (0,0)\}\)), which is exactly the map \(\nu^+\) (resp. \(\nu^-\)) in the statement of Part [P:push-loc4]. The proof of Part [P:push-loc4] is now complete. ◻

Proof. The map \(\Xi_i\) is well-defined and it satisfies [E:Vfun-n431b] since \(\xi_i\) is compatible with the complementary involution and it sends triangles of \(\mathbb{D}_{g,n}\) into triangles of \(\mathbb{D}_{g,n+1}\).

The map \(\Omega\) is well-defined and it satisfies [E:Vfun-n431c] since:

• \(\widehat\sigma\) satisfies the analogous of the properties of Definition 5 on \(\widehat{\mathbb{D}}_{g,n}\) with degeneracy subset \(\mathcal{D}(\widehat\sigma)=\mathcal{D}(\sigma)\cup (\mathbb{D}_{g,n+1}- \mathbb{D}_{g,n})\), as it follows from the definition of \(\widehat\sigma\) and the fact that the unique triangles of \(\widehat{\mathbb{D}}_{g,n}\) not contained in \(\mathbb{D}_{g,n}\) are the ones of the form \(\Delta=[(2;0,\emptyset),(e;h,A),(e;h,A)^c]\), for which Condition 14 reduces to Condition 13 for the complementary pair \(\{(e;h,A),(e;h,A)^c\}\).

• \(\varpi\) is compatible with the complementary involution and it sends triangles of \(\mathbb{D}_{g,n+1}\) into triangles of \(\widehat{\mathbb{D}}_{g,n}\).

Finally, Property [E:Vfun-n431a] follows from the definitions of \(\Xi_i\) and \(\Omega\), together with the fact that \(\varpi\circ \xi_i=\operatorname{id}\). ◻

Proof. Consider the V-stability conditions \(\mathfrak{s}^{\sigma}\in \operatorname{VStab}_{g,n}^\chi\) and \(\mathfrak{s}^{\Omega(\sigma)}\in \operatorname{VStab}_{g,n+1}^\chi\), as in Proposition 10.

First of all, we observe that \(\overline{\mathcal{J}}_{g,n+1}(\Omega(\sigma))\subset ^{o}\operatorname{TF}_{g,n+1}^\chi\), since if \(E\) is an exceptional component of \(C\in \overline{\mathcal{M}}_{g,n+1}(k)\) then \(\operatorname{type}_C(E)=(2;0,\{n+1\})\) or \((1;0,\{j,n+1\})\) for some \(1\leq j \leq n\), and hence \[\mathfrak{s}^{\Omega(\sigma)}(E)=\Omega(\sigma)(\operatorname{type}_C(E))=0 \text{ and } \mathfrak{s}^{\Omega(\sigma)}(E^c)=\Omega(\sigma)(\operatorname{type}_C(E^c))=\chi,\] by definition of \(\Omega(\sigma)\).

Therefore, it remains to show that for any geometric point \((C,I)\) of \(^{o}\operatorname{TF}_{g,n+1}^\chi\), we have that \[\label{E:2-semistab} \begin{align} & I \text{ is } \mathfrak{s}^{\Omega(\sigma)}(C)-\text{semistable, i.e.} \\ & \chi(I_Z)\geq \Omega(\sigma)(\operatorname{type}_C(Z)) \text{ for any } Z\in \operatorname{BCon}(C) \end{align} \Leftrightarrow \begin{align} & \operatorname{st}_*(I) \text{ is } \mathfrak{s}^{\sigma}(C^{\operatorname{st}})-\text{semistable, i.e. } \\ & \chi(\operatorname{st}_*(I)_Y)\geq \sigma(\operatorname{type}_{C^{\operatorname{st}}}(Y)) \text{ for any } Y\in \operatorname{BCon}(C^{\operatorname{st}}). \end{align}\tag{87}\]

The biconnected subcurves of \(C\) and of its stabilization \(C^{\operatorname{st}}\), and their types, are related by the following commutative diagram with surjective horizontal arrows \[\label{E:2BCon} \begin{tikzcd} Z \dar[maps to, rr] && \underline{\operatorname{st}}(Z):=\operatorname{st}(Z) \\ \operatorname{BCon}(C)-\{E,E^c\} \arrow[rr, twoheadrightarrow, "\underline{\operatorname{st}}"] \arrow[d, "\operatorname{type}_C(-)"']&& \operatorname{BCon}(C^{\operatorname{st}}) \arrow[d, "\operatorname{type}_{C^{\operatorname{st}}}(-)"] \\ \varpi^{-1}(\mathbb{D}_{g,n}) \arrow[rr, twoheadrightarrow, "\varpi"]&& \mathbb{D}_{g,n} \end{tikzcd}\tag{88}\] where \(E\) is the exceptional component of \(C\) (if there is one) or \(\emptyset\) (if \(\operatorname{st}\) is an isomorphism). Moreover, the fibers of the map \(\underline{\operatorname{st}}\) are given as follows \[\underline{\operatorname{st}}^{-1}(Y)= \begin{cases} \{\operatorname{st}^{-1}(Y)\} & \text{ if either } E=\emptyset \text{ or } \operatorname{st}(E)\not\in Y\cap Y^c,\\ \{\operatorname{st}^{-1}(Y), \operatorname{st}^{-1}(Y)-E\} & \text{ if } \operatorname{st}(E)\in Y\cap Y^c.\\ \end{cases}\] Observe now that, since \((C,I)\in ^{o}\operatorname{TF}_{g,n+1}^\chi\) by assumption, the sheaf \(I\) satisfies \(\chi(I_E)\geq 0=\Omega(\sigma)(\operatorname{type}_C(E))\) and \(\chi(I_{E^c})\geq \chi=\Omega(\sigma)(\operatorname{type}_C(E^c))\), so that the left hand side of 87 holds true for \(E\) and \(E^c\). Therefore, the equivalence in 87 follows from the surjectivity of \(\underline{\operatorname{st}}\) and the following

Claim: For any \(Y\in \operatorname{BCon}(C^{\operatorname{st}})\), we have that \[\chi(\operatorname{st}_*(I)_Y)=\min_{Z\in \underline{\operatorname{st}}^{-1}(Y)}\{\chi(I_Z)\}.\]

Indeed, the Claim is obvious if either \(E=\emptyset\), i.e. \(\operatorname{st}\) is an isomorphism, or \(\operatorname{st}(E)\not \in Y\cap Y^c\). Suppose now that \(\operatorname{st}(E)\in Y\cap Y^c\), so that \(\underline{\operatorname{st}}^{-1}(Y)=\{\operatorname{st}^{-1}(Y)-E:=\widetilde{Y},\operatorname{st}^{-1}(Y)=\widetilde{Y}\cup E\}\). Denote by \(N\) the node in between \(E\) and \(\widetilde{Y}\) and set \[\delta_N(I)= \begin{cases} 0 &\text{ if } I \text{ is free at } N,\\ 1 &\text{ if } I \text{ is not free at } N.\\ \end{cases}\] We will distinguish two exhaustive (but not mutually exclusive) cases:

\(\bullet\) Case I: \(\chi(I_E)+\delta_N(I)\geq 1\).

In this case, we have that \[\label{E:Iineq} \chi(I_{\widetilde{Y}\cup E})=\chi(I_{\widetilde{Y}})+\chi(I_E)+\delta_N(I)-1\geq \chi(I_{\widetilde{Y}}).\tag{89}\] We will show that \[\label{E:push-res1} \operatorname{st}_*(I)_Y=\operatorname{st}_*(I_{\widetilde{Y}}),\tag{90}\] which, together with the fact that \(R^1\operatorname{st}_*(I_{\widetilde{Y}})=0\) (since \(R^1\operatorname{st}_*(I)=0\)), implies the desired equality \[\chi(\operatorname{st}_*(I)_Y)=\chi(\operatorname{st}_*(I_{\widetilde{Y}}))=\chi(I_{\widetilde{Y}})=\min_{Z\in \underline{\operatorname{st}}^{-1}(Y)}\{\chi(I_Z)\}.\] Consider the exact sequence (see 3 ) \[\label{E:exa-se1} 0\to \leftindex_{(\widetilde{Y})^c}I\to I \to I_{\widetilde{Y}}\to 0.\tag{91}\] Since \(\leftindex_{(\widetilde{Y})^c}I=I_{(\widetilde{Y})^c}(-(\widetilde{Y}\cap \widetilde{Y}^c\cap \operatorname{NF}(I)^c)\) by 5 , we have that \[(\leftindex_{(\widetilde{Y})^c}I)_E= \begin{cases} I_E(-N) & \text{ if } I \text{ is free at } N,\\ I_E & \text{ if } I \text{ is not free at } N.\\ \end{cases}\] In any case, our assumption is equivalent to the fact \(\chi((\leftindex_{(\widetilde{Y})^c}I)_E)\geq 0\), which then implies that \(R^1\operatorname{st}_*(\leftindex_{(\widetilde{Y})^c}I)=0\) by 65 . By taking the push-forward via \(\operatorname{st}\) of the exact sequence in 91 and using the above vanishing, we obtain the surjection \[\operatorname{st}_*(I) \twoheadrightarrow \operatorname{st}_*(I_{\widetilde{Y}}).\] Therefore, using that \(\operatorname{st}_*(I_{\widetilde{Y}})\) is also torsion-free because \(\operatorname{st}_{|\widetilde{Y}}\) is an isomorphism, we deduce that 90 holds.

\(\bullet\) Case II: \(\chi(I_E)+\delta_N(I)\leq 1\).

In this case, we have that \[\label{E:IIineq} \chi(I_{\widetilde{Y}\cup E})=\chi(I_{\widetilde{Y}})+\chi(I_E)+\delta_N(I)-1\leq \chi(I_{\widetilde{Y}}).\tag{92}\] We will show that \[\label{E:push-res2} \operatorname{st}_*(I)_Y=\operatorname{st}_*(I_{\widetilde{Y}\cup E}),\tag{93}\] which, together with the fact that \(R^1\operatorname{st}_*(I_{\widetilde{Y}\cup E})=0\) (since \(R^1\operatorname{st}_*(I)=0\)), will imply the desired equality \[\chi(\operatorname{st}_*(I)_Y)=\chi(\operatorname{st}_*(I_{\widetilde{Y}\cup E}))=\chi(I_{\widetilde{Y}\cup E})=\min_{Z\in \underline{\operatorname{st}}^{-1}(Y)}\{\chi(I_Z)\}.\] Consider the exact sequence (see 3 ) \[\label{E:exa-se2} 0\to \leftindex_{(\widetilde{Y}\cup E)^c}I\to I \to I_{\widetilde{Y}\cup E}\to 0.\tag{94}\] Since the restriction of \(\operatorname{st}\) to \((\widetilde{Y}\cup E)^c\) is an isomorphism, we have that \(R^1\operatorname{st}_*(\leftindex_{(\widetilde{Y}\cup E)^c}I)=0\), which then implies that we have a surjection \[\label{E:suj-push} \operatorname{st}_*(I) \twoheadrightarrow \operatorname{st}_*(I_{\widetilde{Y}\cup E}).\tag{95}\] We now apply our assumption in order to get \[\chi(I_{\widetilde{Y}\cup E})=\chi(I_{\widetilde{Y}})+\chi(I_E)+\delta_N(I)-1\leq \chi(I_{\widetilde{Y}}).\] Then we apply 67 in order to deduce that \(\operatorname{st}_*(I_{\widetilde{Y}\cup E})\) is torsion-free, which together with 95 implies 93 . ◻

Proof. It follows by combining Theorem 23 with Proposition 28. ◻

Proof. In order to construct a graph \(G\) as in Part [L:graph1] (so \(r=0\)), we distinguish two cases:

1. If \(e\) is even, then \(G\) is the graph having vertices \(\{v_1,\ldots, v_q\}\), each of genus \(h\), and having \(e/2\) edges in between \(v_i\) and \(v_{i+1}\) for every \(i=1,\ldots,q\) (with the cyclic convention that \(v_{q+1}=v_1\)).

2. If \(e\) is odd, \(q\) must be even. Then \(G\) is the graph having vertices \(\{v_1,\ldots, v_q\}\), each of genus \(h\), and having \((e-1)/2\) edges in between \(v_i\) and \(v_{i+1}\) for every \(i=1,\ldots,q\) (with the cyclic convention that \(v_{q+1}=v_1\)) and \(1\) edge in between \(v_i\) and \(v_{i+q/2}\) for every \(i=1,\ldots,q/2\).

In order to construct a graph \(G\) as in Part [L:graph2], we distinguish three cases:

1. If \(e\) is even (which implies that \(\delta\) and \(r\) are also even), then \(G\) is the graph having vertices \(\{w, v_1,\ldots, v_q\}\), with each \(v_i\) of genus \(h\), and \(w\) of genus \(r/2>0\), and having \(e/2\) edges in between \(v_i\) and \(v_{i+1}\) for every \(i=1,\ldots,q-1\), with \((e/2-1)\) edges in between \(v_1\) and \(v_q\), and with \(w\) connected with \(1\) edge to \(v_1\) and with \(1\) edge to \(v_q\).

2. If \(e\) is odd and \(r\) is even (which then implies that \(q\) is even), then \(G\) is the graph having vertices \(\{w, v_1,\ldots, v_q\}\), with each \(v_i\) of genus \(h\) and \(w\) of genus \(r/2>0\), and having \((e-1)/2\) edges in between \(v_i\) and \(v_{i+1}\) for every \(i=1,\ldots,q-1\), with \((e-3)/2\) edges in between \(v_1\) and \(v_q\), with \(1\) edge in between \(v_i\) and \(v_{i+q/2}\) for every \(i=1,\ldots,q/2\), and with \(w\) connected with \(1\) edge to \(v_1\) and with \(1\) edge to \(v_q\).

3. If \(e\) is odd and \(r\) is odd (which then implies that \(q\) is also odd), then \(G\) is the graph having vertices \(\{w, v_1,\ldots, v_q\}\), with each \(v_i\) of genus \(h\) and \(w\) of genus \((r-1)/2\), and having \((e-1)/2\) edges in between \(v_i\) and \(v_{i+1}\) for every \(i=1,\ldots,q-1\), with \((e-3)/2\) edges in between \(v_1\) and \(v_q\), with \(1\) edge in between \(v_i\) and \(v_{i+(q+1)/2}\) for every \(i=1,\ldots,(q-1)/2\), and with \(w\) connected with a single edge to \(v_1\), \(v_{(q+1)/2}\) and \(v_q\).

 ◻

Proof of Theorem 30. It is enough to prove that \(\operatorname{^{ns}\Sigma}_g^\chi\) consists of one element, which then must be \(\sigma_g^{\chi, ns}\). We will show that the value of any \(\sigma \in \operatorname{^{ns}\Sigma}_g^\chi\) on a given \((e;h)\in \mathbb{D}_{g}^{ns}\) is uniquely determined by \(\delta(e;h):=2h-2+e\).

Using 13 , it is enough to prove that the degeneracy set \(\operatorname{^{ns}\mathcal{D}}(\sigma)\) is independent of \(\sigma\) and that the value \(\sigma(e;h)\) depends only on \(\delta(e;h)\) for any element \((e;h)\in \mathbb{D}_{g}^{ns}\) such that \(\delta(e;h)\leq g-1\). Write \[2g-2=q\cdot \delta(e;h)+r \text{ with } 0\leq r <\delta(e;h).\] Note that \(q\geq 2\) since \(\delta(e;h)\leq g-1\).

We will now distinguish two cases:

Case I: \(r=0\), i.e. \(\delta(e;h)\) divides \(2g-2\).

By applying Lemma 9[L:graph1], we find a stable Hamiltonian graph of genus \(g\) with \(q\) vertices of type \((e;h)\). Now we apply iteratively \((q-2)\)-times Remark 8 along the Hamiltonian cycle \(\{v_1,\ldots, v_q\}\) of \(G\), and then apply 13 , to obtain \[\label{E:poly1} q\sigma(e;h)-\chi\in \{0,\ldots, q-1\} \text{ with } 0 \text{ occurring if and only if } (e;h)\in \operatorname{^{ns}\mathcal{D}}(\sigma).\tag{99}\] This shows that the value \(\sigma(e;h)\) depends only on \(q\) (and hence on \(\delta(e;h)\)), and that \((e;h)\in \operatorname{^{ns}\mathcal{D}}(\sigma)\) if and only if \(q\) divides \(\chi\) (which shows that this property is independent of \(\sigma\)).

Case II: \(r>0\).

We will proceed by induction on \(\delta(e;h)\), the base of the induction is the case \(\delta(e;h)=1\), which is covered in Case I.

By applying Lemma 9[L:graph2], we find a stable Hamiltonian graph of genus \(g\) with \(q\) vertices of type \((e;h)\) and one vertex of type \[(f;k):= \begin{cases} (2;r/2) & \text{ if } r \text{ is even,}\\ (3;(r-1)/2)& \text{ if } r \text{ is odd.}\\ \end{cases}\]

Now we apply iteratively \((q-2)\)-times Remark 8 along the Hamiltonian cycle \(\{v_1,\ldots, v_q, w\}\) of \(G\), and then apply 13 , to obtain \[\label{E:poly2} q\sigma(e;h)+\sigma(f;k)-\chi\in \begin{cases} \{0,\ldots,q-1\} & \text{ if } (f;k)\in \operatorname{^{ns}\mathcal{D}}(\sigma)\\ &\text{ with } 0 \text{ occurring if and only if } (e;h)\in \operatorname{^{ns}\mathcal{D}}(\sigma),\\ \{1,\ldots,q\} & \text{ if } (f;k)\not \in \operatorname{^{ns}\mathcal{D}}(\sigma).\\ \end{cases}\tag{100}\] Since \(\delta(f;k)=r<\delta(e;h)\), we can apply either Case I (if \(r\) divides \(2g-2\)) or the induction hypothesis (if \(r\) does not divide \(2g-2\)) to deduce that \(\sigma(f;k)\) depends only on \(r\), and that the property that \((f;k)\) belongs to \(\operatorname{^{ns}\mathcal{D}}(\sigma)\) is independent of \(\sigma\). Using this, Equation 100 implies that the value \(\sigma(e;h)\) depends only on \(\delta(e;h)\), and that the property that \((e;h)\) belongs to \(\operatorname{^{ns}\mathcal{D}}(\sigma)\) is independent of \(\sigma\). ◻

Proof. This follows by combing Theorem 30 with Lemma 4 and Theorem 21. ◻

Proof. Part [C:iso-n02]: Theorem 21 implies that \(\overline{J}_g(\sigma_1)\) is isomorphic to \(\overline{J}_g(\sigma_2)\) over \(\overline{\mathcal{M}}_g\) if and only if \(\sigma_1^{ns}\) and \(\sigma_2^{ns}\) lie in the same orbit for the action of \(\operatorname{PR}_g\). Theorem 30 implies that \(\sigma_i^{ns}=\sigma_g^{\chi_i,ns}\) for \(i=1,2\). We conclude by observing that the group \(\operatorname{PR}_g\) is isomorphic to \(\mathbb{Z}\langle \omega_{\pi}\rangle \rtimes (\mathbb{Z}/2\mathbb{Z})\langle \iota\rangle\) (see 36 and the discussion following it) and the action of \(\operatorname{PR}_g\) on \(\Sigma_g\) is such that (by Lemma 5) \[{ \left\{\begin{align} & \omega_{\pi}\cdot \sigma_g^\chi=\sigma_g^{\chi+2g-2},\\ & \iota\cdot \sigma_{\pi}^\chi=\sigma_{\pi}^{-\chi}. \end{align}\right.}\]

Part [C:iso-n01]: Corollary 1 implies that \(\overline{\mathcal{J}}_g(\sigma_1)\) is isomorphic to \(\overline{\mathcal{J}}_g(\sigma_2)\) over \(\overline{\mathcal{M}}_g\) if and only if \(\sigma_1\) and \(\sigma_2\) lie in the same orbit for the action of \(\widetilde{\operatorname{PR}}_g\). In turn, Lemma 5 implies that \(\sigma_1\) and \(\sigma_2\) lie in the same orbit for the action of \(\widetilde{\operatorname{PR}}_g\) if and only if \(\mathcal{D}(\sigma_1^s)=\mathcal{D}(\sigma_2^s)\) and \(\sigma_1^{ns}\) and \(\sigma_2^{ns}\) lie in the same orbit for the action of \(\operatorname{PR}_g\). We conclude using what observed in the proof of Part [C:iso-n02]. ◻

Proof. Part [C:fineMg1]: first of all, by 101 and 98 , we get that \[\label{E:Cap-nf} \overline{\mathcal{J}}_{g}^{Cap, \chi}\text{ is non-fine } \Longleftrightarrow \frac{\chi}{2g-2}(2h-2+e)\in \mathbb{Z}\text{ for some } (e;h)\in \mathbb{D}_g.\tag{102}\] Since \(0<2h-2+e<2g-2\) for any \((e;h)\in \mathbb{D}_g\) and all the numbers in the interval \((0,2g-2)\) are of the form \(2h-2+e\) for some \((e;h)\in \mathbb{D}_g\), Condition 102 implies that \[\label{E:Cap-nf2} \overline{\mathcal{J}}_{g}^{Cap, \chi}\text{ is fine } \Longleftrightarrow \gcd(\chi, 2g-2)= 1.\tag{103}\] This shows the if part of the first assertion.

Conversely, if \(\gcd(\chi, 2g-2)=k\geq 2\) then \((\frac{2g-2}{k}+2; 0)\in \mathcal{D}(\sigma_g^{\chi})\) by 98 , Hence, since \((\frac{2g-2}{k}+2; 0)\in \mathbb{D}_g^{ns}\), Theorem 3[C:class-n01] implies that \((\frac{2g-2}{k}+2; 0)\) belongs to \(\mathcal{D}(\sigma)\) for any \(\sigma \in \Sigma_g^\chi\), which implies that there are no fine compactified Jacobians over \(\overline{\mathcal{M}}_{g}\) of characteristic \(\chi\). This proves the only if part of the first assertion.

The second assertion follows from Corollary 4[C:iso-n01].

Part [C:fineMg2]: the \(\operatorname{\mathbb{G}_m}\)-gerbe \(\mathcal{J}_g^{\chi}\to J_g^{\chi}=\mathcal{J}_g^{\chi}\fatslash \operatorname{\mathbb{G}_m}\) is trivial if and only if \(\gcd(\chi, 2g-2)=1\) by [47]. Therefore, if \(\gcd(\chi, 2g-2)\neq 1\) there cannot be any tautological sheaf on \(J_g^\chi\times_{{\mathcal{M}}_{g}} {\mathcal{C}}_{g}\to J_g^\chi\) and hence neither on \(\overline{J}_{g}^{Cap, \chi}\times_{\overline{\mathcal{M}}_{g}} \overline{\mathcal{C}}_{g}\to \overline{J}_{g}^{Cap, \chi}\). Conversely, if \(\gcd(\chi, 2g-2)=1\) then \(\overline{J}_g^{Cap,\chi}=\overline{\mathcal{J}}_g^{Cap,\chi}\fatslash \operatorname{\mathbb{G}_m}\) by part [C:fineMg1] and the \(\operatorname{\mathbb{G}_m}\)-gerbe \(\overline{\mathcal{J}}_g^{Cap,\chi}\to \overline{\mathcal{J}}_g^{Cap,\chi}\fatslash \operatorname{\mathbb{G}_m}\) is trivial by what recalled above (because being trivial is Zariski locally on the base). Hence, the universal sheaf \(\mathcal{I}_{g,n}\) on \(\overline{\mathcal{J}}_g^{Cap,\chi}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}\) descends (non-uniquely) to a tautological sheaf \(\mathcal{I}_{g,n}^{\operatorname{taut}}\) on \(\overline{J}_g^{Cap,\chi}\times_{\overline{\mathcal{M}}_{g}}\overline{\mathcal{C}}_{g}\). ◻

Proof. Part [P:Deg-map1] follows immediately from the explicit description of the action of \(\widetilde{\operatorname{PR}}_{g,n}\) on \(\Sigma_{g,n}\), see Remark 15. Part [P:Deg-map4] follows from Remark 9.

Part [P:Deg-map2]: consider two V-functions \(\sigma^1,\sigma^2\in \Sigma_{g,n}\) such that \(\sigma^1\geq \sigma^2\) (which implies that they have the same characteristic \(|\sigma^1|=|\sigma^2|:=\chi\)) and let us show that \(D^1:=\mathcal{D}(\sigma^1)\geq D^2:=\mathcal{D}(\sigma^2)\). First of all, \(D^1\subseteq D^2\) by Remark 9. We now show that the subset \[E:=\{(e;h,A)\in D^2-D^1\: : \; \sigma_1(e;h,A)=\sigma_2(e;h,A)+1\}.\] is a witness for the relation \(D^1\geq D^2\), i.e. it satisfies the conditions of Definition 10[D:degsub2]:

\(\bullet\) Condition [D:degsub2a] follows from Remark 9.

\(\bullet\) Consider a triangle \(\Delta=[(e_1;h_1,A_1), (e_2;h_2,A_2), (e_3;h_3,A_3)]\) such that \((e_1;h_1,A_1)\in D^1\subseteq D^2\) and \((e_2;h_2,A_2), (e_3;h_3,A_3)\in D^2-D^1\). By applying 14 to \(\sigma^1\) and \(\sigma^2\) and the triangle \(\Delta\) we get that \[{ \left\{\begin{align} & \sum_{i=1}^3\sigma^1(e_i;h_i,A_i)-\chi=1,\\ & \sum_{i=1}^3\sigma^2(e_i;h_i,A_i)-\chi=0. \end{align}\right.}\] This implies that \(E\) contains exactly one among \((e_2;h_2,A_2)\) and \((e_3;h_3,A_3)\), and hence condition [D:degsub2b] holds.

\(\bullet\) Consider a triangle \(\Delta=[(e_1;h_1,A_1), (e_2;h_2,A_2), (e_3;h_3,A_3)]\) such that \((e_i;h_i,A_i)\in D^1\subseteq D^2\) for any \(1\leq i \leq 3\). By applying 14 to \(\sigma^1\) and \(\sigma^2\) and the triangle \(\Delta\) we get that \[{ \left\{\begin{align} & \sum_{i=1}^3\sigma^1(e_i;h_i,A_i)-\chi\in \{1,2\},\\ & \sum_{i=1}^3\sigma^2(e_i;h_i,A_i)-\chi=0. \end{align}\right.}\] This implies that \(E\) contains either \(1\) or \(2\) elements of \(\Delta\), and hence Condition [D:degsub2c] holds.

Part [P:Deg-map3]: consider two degeneracy subsets \(D^1,D^2\in \operatorname{Deg}_{g,n}\) such that \(D^1\geq D^2\) in \(\operatorname{Deg}_{g,n}\) and a V-function \(\sigma^2\in \Sigma_{g,n}\) such that \(\mathcal{D}(\sigma^2)=D^2\). By Definition 10[D:degsub2], there exists \(E\subset D^2-D^1\) satisfying the conditions of loc. cit. We now consider the function \(\sigma^1:\mathbb{D}_{g,n}\to \mathbb{Z}\) defined by \[\sigma^1(e;h,A):= { \left\{\begin{align} \sigma^2(e;h,A) & \text{ if either } (e;h,A)\in D^1 \text{ or } (e;h,A)\not \in D^2 \text{ or } (e;h,A)\in E^c,\\ \sigma^2(e;h,A)+1 & \text{ if } (e;h,A)\in E. \end{align}\right.}\] We will now show that \(\sigma^1\) is a V-function of characteristic \(\chi:=|\sigma^2|\) with \(\mathcal{D}(\sigma^1)=D^1\), which clearly concludes the proof since \(\sigma^1\geq \sigma^2\) by construction. In order to show this, we will check the conditions of Definition 5.

Equation 13 holds for \(\sigma^1\) by the property in [D:degsub2a] of \(E\) and we also get that the \(\sigma^1\)-degenerate elements are exactly the elements of \(D^1\). Hence, the first condition of Definition 5[E:condUni2] holds true since \(D^1\) is a degeneracy subset.

In order to check Condition 14 , consider a triangle \(\Delta=[(e_1;h_1,A_1), (e_2;h_2,A_2), (e_3;h_3,A_3)]\) and we will distinguish several cases:

\(\bullet\) If either \((e_i;h_i,A_i)\in D^1\) for all \(i\), or \((e_i;h_i,A_i)\not \in D^2\) for all \(i\), or one of the elements of \(\Delta\) belongs to \(D^1\) and the other two do not belong to \(D^2\), then we have that \(\sigma^1(e_i;h_i,A_i)=\sigma^2(e_i;h_i,A_i)\) for every \(i\) and hence Condition 14 for \(\sigma^2\) implies that the same condition holds for \(\sigma^1\).

\(\bullet\) If (up to permuting the indices) we have that \((e_1;h_1,A_1)\in D^2-D^1\) and \((e_2;h_2,A_2),(e_3;h_3,A_3)\not \in D^2\), then we compute \[\sum_{i=1}^3\sigma^1(e_i;h_i,A_i)-\chi=\sum_{i=1}^3\sigma^2(e_i;h_i,A_i)-\chi+|\{(e_1;h_1,A_1)\}\cap E|\in 1+\{0,1\}=\{1,2\}.\]

\(\bullet\) If we have that \((e_i;h_i,A_i)\in D^2-D^1\) for all \(i\), then we compute \[\sum_{i=1}^3\sigma^1(e_i;h_i,A_i)-\chi=\sum_{i=1}^3\sigma^2(e_i;h_i,A_i)-\chi+|\{(e_i;h_i,A_i)\in E\}|\in 0+\{1,2\}=\{1,2\},\] where we have used Condition [D:degsub2c] for \(E\).

\(\bullet\) If (up to permuting the indices) we have that \((e_1;h_1,A_1)\in D^1\) and \((e_2;h_2,A_2),(e_3;h_3,A_3) \in D^2-D^1\), then we compute \[\sum_{i=1}^3\sigma^1(e_i;h_i,A_i)-\chi=\sum_{i=1}^3\sigma^2(e_i;h_i,A_i)-\chi+|\{(e_2;h_2,A_2), (e_3;h_3,A_3)\}\cap E|\in 0+\{0,1\}=\{0,1\},\] where we have used Condition [D:degsub2b] for \(E\).

In each of the above cases, Condition 14 is satisfied for \(\sigma^1\) with respect to the triangle \(\Delta\), and we are done. ◻

Proof. This follows by combining the proofs of Parts [P:Deg-map2] and [P:Deg-map3] of Proposition 33. ◻

Proof. This follows from Proposition 33[P:Deg-map1], together with the explicit action of \(\widetilde{\operatorname{PR}}_{g,n}\) on \(\Sigma\) (see Remark 15). ◻

Proof. Part [P:Dsep1] follows from Lemma 5.

The other parts follows easily from the Definition 8 of \(\operatorname{^{s}\Sigma}_{g,n}^\chi\) and the above definition of \(\operatorname{Deg}_{g,n}\) (using that there are no triangles in \({^{s}\mathbb{D}_{g,n}}\)). ◻

Proof. Part [C:height1] follows from Proposition 33: the inequality \(h(\sigma)\geq h(\mathcal{D}(\sigma))\) follows from the fact that \(\mathcal{D}\) is upper-lifting, the inequality \(h(\sigma)\leq h(\mathcal{D}(\sigma))\) follows from the fact that \(\mathcal{D}\) is order-preserving and conservative.

Part [C:height2] follows from the isomorphisms of Lemma 3 and 105 .

Part [C:height3] follows from Proposition 34[P:Dsep3] and the fact that \(\operatorname{^{s}\operatorname{Deg}}_{g,n}\) is the collection of all subsets of \({^{s}\mathbb{D}_{g,n}}\) satisfying Condition [D:degsub1a] of Definition 10. ◻

Proof. Observe that: \(\sigma\) is uniform if and only if \(\sigma^{ns}\) is uniform by definition, \(\sigma\) is classical if and only if \(\sigma^{ns}\) is classical by Remark 19, and that \(\operatorname{^{ns}\mathcal{D}}(\sigma)=\mathcal{D}(\sigma^{ns})\). Hence, we can assume for the rest of the proof that \(\sigma\in \operatorname{^{ns}\Sigma}_{g,n}^\chi\).

Let us first show the only if implication. According to 24 , the classical non-separating V-functions of characteristic \(\chi\) are given by (for any \(x=(e;h,A)\in {^{ns}\mathbb{D}_{g,1}}\)) \[\label{E:sigma-beta} \sigma_{\beta}(x):=\sigma_{\beta\omega_{\pi}+(\chi-(2g-2)\beta)\Sigma_1}(x)= \begin{cases} \lceil \delta(x)\beta\rceil & \text{ if } x\in {^{ns}\mathbb{D}^e_{g,1}}=\mathbb{D}_{g,1}^{pr}, \\ \lceil \chi-\beta\delta(x^c)\rceil & \text{ if } x\in {^{ns}\mathbb{D}^f_{g,1}}, \end{cases}\tag{121}\] as \(\beta\) varies in \(\mathbb{R}\). The above equation shows that any classical (non-separating) V-function is uniform and that \[\label{E:D-sigbeta} \mathcal{D}(\sigma_{\beta})^{pr}=\left\{x=(e;h)\in \mathbb{D}_{g,1}^{pr}: \: \beta\,\delta(x)\in\mathbb{Z}\right\}.\tag{122}\] Therefore, if \(\beta\not\in \mathbb{Q}\) then \(\mathcal{D}(\sigma_\beta)^{pr}=\emptyset\), which implies that \(\mathcal{D}(\sigma_\beta)=\emptyset\) by Remark 36[R:Deg-Sig-pr1]. Otherwise, we write \(\beta=\frac{p}{q}\) in lowest terms with \(q>0\). Then \(\beta\,\delta(x)\in\mathbb{Z}\) is equivalent to \(q\mid \delta(x)\). If \(q>2g-2\) then again \(\mathcal{D}(\sigma_\beta)=\emptyset\). On the other hand, if \(q\leq 2g-2\) then \(\mathcal{D}(\sigma_\beta)^{pr}=W_q^{pr}\), which implies that \(\mathcal{D}(\sigma_\beta)=W_q\) by Remark 36[R:Deg-Sig-pr1]. This computation also proves the last statement since \[{ \left\{\begin{align} & \emptyset=\mathcal{D}(\sigma_\beta) \text{ for any } \beta\not \in \mathbb{Q}, \\ & W_\delta=\mathcal{D}(\sigma_{1/\delta}) \text{ for } 1\leq \delta\leq 2g-2. \end{align}\right.}\]

We now show the if implication. Let \(\sigma\in\operatorname{^{ns}\Sigma}^\chi_{g,1}\) be a uniform V-function, and denote by \(\sigma(\delta):=\sigma(e;h)\) for each \((e;h)\in\mathbb{D}_{g,1}^{pr}\) such that \(\delta(e;h)=\delta\in [1,2g-2]\). We now distinguish two cases according to \(\mathcal{D}(\sigma)\).

\(\underline{\text{Case I:}}\) \(\operatorname{^{ns}\mathcal{D}}(\sigma)=W_\delta\) for some \(1\leq \delta\leq 2g-2\).

We will prove that (see 121 for the notation) \(\displaystyle \sigma=\sigma_{\frac{\sigma(\delta)}{\delta}}\), which amounts to showing, by 121 and 122 together with Remark 36[R:Deg-Sig-pr2], the following two properties: \[\label{E:betaWdelta1} \sigma(\delta')=\bigg\lceil\frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil \text{ for any } 1\leq \delta'\leq 2g-2.\tag{123}\] \[\label{E:betaWdelta2} \frac{\sigma(\delta)\delta'}{\delta}\in \mathbb{Z}\Longleftrightarrow \delta\vert \delta', \text{ for any } 1\leq \delta'\leq 2g-2.\tag{124}\] Observe that 124 is equivalent to the condition \[\label{E:coprime} \gcd(\sigma(\delta),\delta)=1,\tag{125}\] which we are now going to check. Consider any decomposition \(\delta=k\cdot h\) with \(h,k\in \mathbb{N}_{>0}\) such that \(h\neq 1\) (so that \(k\neq \delta\)). Property 120 and the assumption \(\mathcal{D}(\sigma)=W_{\delta}\) imply that \[{ \left\{\begin{align} & -1\leq \sigma(ik)-\sigma((i-1)k)-\sigma(k)\leq 0 \text{ for any } 2\leq i \leq h-1,\\ & \sigma(\delta)=\sigma(hk)=\sigma((h-1)k)+\sigma(k)-1. \end{align}\right.}\] By summing the above inequalities, we get that \[\label{E:inequa1} -(h-1)\leq \sigma(\delta)-h\sigma(k)\leq -1.\tag{126}\] We deduce that \(h\) does not divide \(\sigma(\delta)\) and, since this is true for all divisors \(1<h\) of \(\delta\), we infer that 125 holds true.

We now check 123 by distinguishing three cases.

\((a)\) If \(\delta'\vert \delta\) then 123 follows from 126 if \(\delta'<\delta\) and it is trivial if \(\delta'=\delta\).

\((b)\) If \(1\leq \delta'<\delta\), we proceed by induction on \(\delta'\). We can assume that \(\delta\geq 2\), for otherwise there is nothing to prove. The base case of the induction is \(\delta'=1\), which follows from Case (a).

We now prove the inductive step. Write \(\delta-\delta'=k\delta'+l\) for some \(k,l\in \mathbb{N}\) such that \(0\leq l<\delta'\). We can assume that \(1\leq l\), for otherwise \(\delta'\vert \delta\) and the result follows from Case (a). Hence, by induction and the definition of \(l\), we have that \[\label{E:indu-l} \sigma(l)=\bigg\lceil\frac{\sigma(\delta)l}{\delta}\bigg\rceil=\bigg\lceil\frac{\sigma(\delta)(\delta-(k+1)\delta')}{\delta}\bigg\rceil=\sigma(\delta)+\bigg\lceil-\frac{\sigma(\delta)(k+1)\delta'}{\delta}\bigg\rceil=\sigma(\delta)-\bigg\lceil\frac{\sigma(\delta)(k+1)\delta'}{\delta}\bigg\rceil+1,\tag{127}\] where in the last equality we have used \[\frac{\sigma(\delta)(k+1)\delta'}{\delta}\not \in \mathbb{Z} \text{ by \eqref{E:betaWdelta2} and } \delta=(k+1)\delta'+l>(k+1)\delta'\] and the property \(\lceil -x\rceil=-\lceil x\rceil +1\) for any \(x\not \in \mathbb{Z}\).

Using this, a repeated application of 120 (together with the assumption that \(\mathcal{D}(\sigma)=W_\delta\)) yields the following \[\label{E:tante-in} { \left\{\begin{align} & \sigma(\delta)-\sigma(\delta')-\sigma(\delta-\delta')=-1,\\ & -1\leq\sigma(\delta-\delta')-\sigma(k\delta')-\sigma(l)\leq 0 \quad \text{ if } k\geq 1,\\ &-(k-1)\leq\sigma(k\delta')-k\sigma(\delta')\leq 0 \quad \text{ if } k\geq 1. \end{align}\right.}\tag{128}\] By summing all the above inequalities, we get the following inequality (note that this also holds true when \(k=0\)) \[\label{E:final-in} -(k+1)\leq \sigma(\delta)-(k+1)\sigma(\delta')-\sigma(l)\leq -1.\tag{129}\] From this inequality and 127 , we deduce that \[\label{E:bounddelta39} { \left\{\begin{align} & \sigma(\delta')\geq \frac{1}{k+1}\bigg(\sigma(\delta)-\sigma(l)+1\bigg)=\frac{1}{k+1}\bigg\lceil\frac{\sigma(\delta)(k+1)\delta'}{\delta}\bigg\rceil\geq \bigg\lceil\frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil-1+\frac{1}{k+1},\\ & \sigma(\delta')\leq\frac{1}{k+1}\bigg(\sigma(\delta)-\sigma(l)+k+1\bigg)=\frac{1}{k+1}\bigg(\bigg\lceil\frac{\sigma(\delta)(k+1)\delta'}{\delta}\bigg\rceil+k\bigg)\leq \bigg\lceil\frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil+1-\frac{1}{k+1}, \end{align}\right.}\tag{130}\] where we have used that \[\lceil x\rceil -1+\frac{1}{h}\leq \frac{\lceil hx\rceil}{h} \leq \lceil x\rceil \quad \text{ for any } x\not \in \mathbb{Z}\text{ and any } h\in \mathbb{N}_{>0}.\] The two inequalities in 130 force the equality \(\displaystyle \sigma(\delta')=\bigg\lceil\frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil\), i.e. 123 holds true for \(\delta'\).

\((c)\) For an arbitrary \(1\leq \delta'\leq 2g-2\), write \(\delta'=k\delta+l\), with \(0\leq l<\delta\) and \(k\geq 0\). If \(l=0\) then we compute \[\sigma(\delta')=\sigma(k\delta)=k\sigma(\delta)= \frac{\sigma(\delta)\delta'}{\delta}= \bigg\lceil \frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil,\] where we have used 120 and the assumption that \(\mathcal{D}(\sigma)=W_\delta\) in the second equality.

Otherwise, if \(1\leq l <\delta\) then we compute \[\begin{align} \sigma(\delta')&=\sigma(k\delta+l)=k\sigma(\delta)+\sigma(l) & \text{ by \eqref{E:sigmapr-prop} using that } \mathcal{D}(\sigma)=W_{\delta}\\ &= k\sigma(\delta)+\bigg\lceil \frac{\sigma(\delta)l}{\delta}\bigg\rceil & \text{ by Case (b) applied to l}\\ &= \bigg\lceil \frac{\sigma(\delta)(k\delta+l)}{\delta}\bigg\rceil=\bigg\lceil \frac{\sigma(\delta)\delta'}{\delta}\bigg\rceil. \end{align}\] which concludes the proof of 123 .

\(\underline{\text{Case II:}}\) \(\operatorname{^{ns}\mathcal{D}}(\sigma)=\emptyset\).

We will show that there exists \(\beta\in \mathbb{R}-\mathbb{Q}\) such that \(\sigma=\sigma_\beta\). The latter amounts to showing, by 121 and 122 together with Remark 36[R:Deg-Sig-pr2], the equality \[\label{E:beta2} \sigma(\delta)=\lceil\beta\delta\rceil, \quad\mathrm{for any}\quad 1\leq\delta\leq2g-2.\tag{131}\] The above condition 131 for \(\beta\) is equivalent to \[\label{E:beta3} \frac{\sigma(\delta)-1}{\delta}<\beta < \frac{\sigma(\delta)}{\delta}, \quad\mathrm{for any}\quad 1\leq\delta\leq2g-2.\tag{132}\] Hence, we have to show that \[\label{E:beta4} \bigcap_{1\leq \delta\leq 2g-2} \left(\frac{\sigma(\delta)-1}{\delta}, \frac{\sigma(\delta)}{\delta}\right)\neq \emptyset,\tag{133}\] where \((a,b)\) denotes as usual the open interval of real numbers larger than \(a\) and smaller than \(b\). Condition 133 is equivalent to showing that the lower bound of each interval is smaller than the upper bound of every other interval, which translates into the following inequalities \[\frac{\sigma(\delta)-1}{\delta}< \frac{\sigma(\delta')}{\delta'} \text{ for any } 1\leq \delta, \delta'\leq 2g-2,\] or equivalently into \[\label{E:beta5} -\delta+1\leq \delta'\sigma(\delta)-\delta\sigma(\delta')\leq \delta'-1 \text{ for any } 1\leq \delta'<\delta\leq 2g-2.\tag{134}\] We will now check 134 by distinguishing two cases:

\((a)\) If \(\delta'\vert \delta\), i.e. \(\delta=k\delta'\) for some \(k\in \mathbb{N}_{>0}\), then the iteration of Property 120 and the assumption \(\mathcal{D}(\sigma)=\emptyset\) imply that \[-(k-1)\leq \sigma(\delta)-k\sigma(\delta')\leq 0.\] By multiplying the above inequality times \(\delta'\) and using that \(\delta'\geq 1\), we obtain 134 in this case.

\((b)\) In the general case \(1\leq \delta'<\delta\), we proceed by induction on the pairs \((\delta'<\delta)\), lexicographically ordered. The base case of the induction is the case \((\delta'=1< \delta)\) where the result follows from Case (a). Write \(\delta=k\delta'+l\) with \(k\in \mathbb{N}_{>0}\) and \(0\leq l <\delta'\). If \(l=0\), then \(\delta'\vert \delta\) and the result follows from Case (a). So we can assume that \(1\leq l <\delta'\).

Again, Property 120 and the assumption \(\mathcal{D}(\sigma)=\emptyset\) imply that \[\label{E:ineq-bet1} -k\leq \sigma(\delta)-k\sigma(\delta')-\sigma(l)\leq 0.\tag{135}\] On the other hand, the induction assumption applied to the pair \((l<\delta')\) (which is lexicographically smaller than \((\delta'< \delta)\)) implies \[\label{E:ineq-bet2} -\delta'+1\leq l\sigma(\delta')-\delta'\sigma(l)\leq l-1.\tag{136}\] By multiplying 135 for \(\delta'\) and subtracting 136 , we deduce that 134 holds true. ◻

Proof. If \(D=W_{\delta(D)}\) then by Theorem 37 \(D\) is classical, and hence it is realizable.

Conversely, assume that \(D\) is realizable, i.e. \(D=\mathcal{D}(\sigma)\) for some \(\sigma\in \operatorname{^{ns}\Sigma}_{g,1}^\chi\), and let us show that \(D=W_{\delta(D)}\). By Remark 36[R:Deg-Sig-pr1], it is enough to show that \(D^{pr}=W_{\delta}^{pr}\). We divide the proof in three steps.

Step I: If \(x=(e;h)\in \mathbb{D}_{g,1}^{pr}\) is such that \(\delta(x)=\delta(D)\) then \(x\in D^{pr}\).

Indeed, by contradiction suppose that \(x\not \in D^{pr}\). By definition of \(\delta(D)\), there exists \(\widetilde{x}=(\widetilde{e};\widetilde{h})\in D^{pr}\) with \(\delta(\widetilde{x})=\delta(D)\). Consider the two compositions \[(e;h)\circ_{e-1} (e;h)=(2;\delta(D))=(\widetilde{e};\widetilde{h})\circ_{\widetilde{e}-1} (\widetilde{e};\widetilde{h}).\] Since \(\widetilde{x}\in D^{pr}\), we have that \((2;\delta(D))\in D^{pr}\) by 118 . We now apply 120 to the two compositions above in order to get \[{ \left\{\begin{align} \sigma(2;\delta(D))-2\sigma(\widetilde{x})=0,\\ \sigma(2;\delta(D))-2\sigma(x)=-1. \end{align}\right.}\] This is absurd since the first equation implies that \(\sigma(2;\delta(D))\) is even and the second equation implies that \(\sigma(2;\delta(D))\) is odd.

Step II: \(W_{\delta(D)}^{pr}\subset D^{pr}\).

Indeed, we will show, by induction on \(k\geq 1\), that for any \(x\in \mathbb{D}_{g,1}^{pr}\) we have \[\delta(x)=k\delta(D)\Rightarrow x\in D^{pr}.\] The base case \(k=1\) of the induction follows from Step I. Assume that \(k\geq 2\). We claim that \(x=(e;h)> (\delta(D)+2;0)\) since \(h\geq 0\) and \[2h+2e\geq (2h+e-2)+4=k\delta(D)+4\geq 2\delta(D)+4>2(r+2),\] where we used that \(e\geq 2\) and that \(\delta(x)=k\delta(D)\geq 2\delta(D)\). Therefore, 115 implies that we can write \[\label{E:x-tria} x=(e;h)=(\delta(D)+2;0)\circ_i \widetilde{x} \text{ for some } \widetilde{x} \in \mathbb{D}_{g,1}^{pr}\text{ and some } i.\tag{138}\] By 111 and 139 , we get that \(\delta(\widetilde{x})=\delta(x)-\delta(\delta(D)+2;0)=(k-1)\delta(D)\), which implies that \(\widetilde{x}\in D^{pr}\) by our induction hypothesis. By applying 118 to the composition 138 , we get that \(x\in D^{pr}\), as required.

Step III: \(W_{\delta(D)}^{pr}\supseteq D^{pr}\).

Indeed, by contradiction, assume that there exists \(x=(e;h)\in D^{pr}-W_{\delta(D)}^{pr}\). By definition of \(\delta(D)\), we must have \(\delta(x)\geq \delta(D)\). Then, using that \(x\not \in W_{\delta(D)}^{pr}\), we can write \[\label{E:div-delta} \delta(x)=k\delta(D)+r \text{ for some } 0<r<\delta(D) \text{ and } k\geq 1.\tag{139}\] We now claim that \(x=(e;h)> (r+2;0)\) since \(h\geq 0\) and \[2h+2e\geq (2h+e-2)+4=k\delta(D)+r+4\geq \delta(D)+r+4>2r+4,\] where we used that \(e\geq 2\) and 139 . Therefore, 115 implies that we can write \[\label{E:trian-x} x=(e;h)=(r+2;0)\circ_i \widetilde{x} \text{ for some } \widetilde{x} \in \mathbb{D}_{g,1}^{pr}\text{ and some } i.\tag{140}\] By 111 and 139 , we get that \(\delta(\widetilde{x})=\delta(x)-\delta(r+2;0)=k\delta(D)\), which implies that \(\widetilde{x}\in W_{\delta(D)}^{pr}\subseteq D^{pr}\) by Step II. By applying 118 to the composition 140 , we get that \((r+2;0)\in D^{pr}\) which is an absurd since \(\delta(r+2;0)=r<\delta(D)\). ◻

Proof. Part [P:deg-grandi2]: let \(D\in \operatorname{^{ns}\operatorname{Deg}}_{g,1}\) with \(\delta(D)\geq g\). Since \(D=D^{pr}\sqcup \widetilde{D^{pr}}\) by 117 , it is enough to prove that \(D^{pr}\subseteq \mathbb{D}_{g,1}^{\geq g}\) is an antichain. Indeed, by contradiction, suppose that \(x_1,x_2\in D^{pr}\) are such that \(x_1< x_2\). Then 115 gives that \[x_2=x_1\circ_i y \text{ for some } y\in \mathbb{D}_{g,1}^{pr}.\] Applying 118 to the above composition, we deduce that \(y\in D^{pr}\). Moreover, 111 and the assumption \(\delta(D)\geq g\) imply that \[\delta(y)=\delta(x_2)-\delta(x_1)\leq 2g-2-g=g-2<\delta(D),\] which is an absurd by the definition of \(\delta(D)\).

Part [P:deg-grandi1]: let \(A\) be an antichain of \(\mathbb{D}_{g,1}^{\geq g}\). We claim that \[\label{E:inter-A} |[x_1,x_2,x_1\circ_i x_2]\cap A|\leq 1,\tag{141}\] for any two elements \(x_1,x_2\in \mathbb{D}_{g,1}^{pr}\) such that \(x_1\circ_i x_2\) is well-defined. Indeed, since the composition \(x_1\circ_i x_2\) is well-defined, then at least one of the two elements \(\{x_1,x_2\}\), say \(x_1\), does not belong to \(\mathbb{D}_{g,1}^{\geq g}\) and hence does not belong to \(A\). On the other hand, since \(x_2< x_1\circ_i x_2\) by 115 and \(A\) is an antichain, it follows that \(A\) cannot contain both \(x_2\) and \(x_1\circ_i x_2\), and 141 is proved. Property 141 implies that \(A\) satisfies 118 and, hence, \(D(A)=A\sqcup \widetilde{A}\) belongs to \(\operatorname{^{ns}\operatorname{Deg}}_{g,1}\) by Remark 36[R:Deg-Sig-pr1]. Moreover, we have that \(\delta(D(A))\geq g\) since \(D(A)^{pr}=A\subset \mathbb{D}_{g,1}^{\geq g}\).

It remains to prove that any such \(D(A)\) is realizable. Consider the function \[\label{E:tauA} \begin{align} \tau_A: \mathbb{D}_{g,1}^{pr}& \longrightarrow \mathbb{Z}\\ x &\mapsto \begin{cases} 2 & \text{ if } x>y \text{ for some } y\in A,\\ 1 & \text{ otherwise.} \end{cases} \end{align}\tag{142}\] Consider now two elements \(x_1, x_2\in \mathbb{D}_{g,1}^{pr}\) such that there exists \(x_1\circ_i x_2\in \mathbb{D}_{g,1}^{pr}\). One of the two elements \(\{x_1,x_2\}\), say \(x_1\), must have log-canonical degree less than or equal to \(g-1\), which then implies that \(\tau_A(x_1)=1\) because \(x_1\) cannot dominate any element of \(A\) since \(A\subseteq \mathbb{D}_{g,1}^{\geq g}\). We now distinguish the following cases:

\(\bullet\) If \(x_2\in A\) then we have that \(\tau_A(x_2)=1\) because \(x_2\) cannot strictly dominate another element of \(A\) since \(A\) is an antichain and \(\tau_A(x_1\circ_i x_2)=2\) because \(x_1\circ_i x_2> x_2\in A\) by 115 .

\(\bullet\) If \(x_1\circ_i x_2\in A\) (which then implies that \(x_2\not \in A\) since \(x_1\circ_i x_2> x_2\in A\) and \(A\) is an antichain) then we have that \(\tau_A(x_2)=\tau_A(x_1\circ_i x_2)=1\) because neither \(x_1\circ_i x_2\) nor \(x_2\) (which is smaller than \(x_1\circ_i x_2\)) can strictly dominate an element of \(A\) since \(A\) is an antichain.

\(\bullet\) If \(x_2,x_1\circ_i x_2\not\in A\) then we must have that \((\tau_A(x_2),\tau_A(x_1\circ_i x_2))\neq (2,1)\) because if \(x_2\) strictly dominate an element of \(A\) then the same is true for \(x_1\circ_i x_2\) (which is bigger then \(x_2\)). Hence, we must have that \((\tau_A(x_2),\tau_A(x_1\circ_i x_2))=(1,1), (1,2),(2,2)\).

By putting all the cases together, we deduce that \[\label{E:tauA-prop} \tau_A(x_1\circ_i x_2)-\tau_A(x_1)-\tau_A(x_2)= \begin{cases} 0 & \text{ if either } x_1\in A \text{ or } x_2\in A, \\ -1 & \text{ if } x_1, x_2\not \in A \text{ and } x_1\circ_i x_2\in A, \\ \{0,-1\} & \text{ otherwise.} \end{cases}\tag{143}\] We can now extend \(\tau_A\) to a function \[\begin{align} \sigma_A: {^{ns}\mathbb{D}_{g,1}}& \longrightarrow \mathbb{Z}\\ x & \mapsto \begin{cases} \tau_A(x) & \text{ if } x\in \mathbb{D}_{g,1}^{pr}, \\ \chi+1-\tau_A(x^c) & \text{ if } x\not \in \mathbb{D}_{g,1}^{pr}\text{ and } x^c\not \in A,\\ \chi-\tau_A(x^c) & \text{ if } x\not \in \mathbb{D}_{g,1}^{pr}\text{ and } x^c\in A.\\ \end{cases} \end{align}\] Using that \(\tau_A\) satisfies 143 with respect to \(A=D(A)^{pr}\), we get that \(\sigma_A\in \operatorname{^{ns}\Sigma}^\chi_{g,1}\) with \(\mathcal{D}(\sigma_A)=D(A)\) by Remark 36[R:Deg-Sig-pr2]. ◻

Proof. The description of the elements of \(\operatorname{^{ns}\operatorname{Deg}}_{g,1}^{re}\) follows from Propositions 38 and 39. It remains to prove that the unique non-trivial order relations are the ones given in [T:pos-Degn1a] and [T:pos-Degn1b].

Let us, first, prove Part [T:pos-Degn1b]. First of all, if \(A,B\) are two antichains of \(\mathbb{D}_{g,1}^{\geq g}\) such that \(A\supsetneq B\), then \(D(A)\supsetneq D(B)\). Then the subsets \(E\) of \(D(A)-D(B)\) that satisfy Condition [D:degsub2a] of Definition 10 are all the ones of the form \(E:=\mathcal{E}\sqcup \widetilde{\mathcal{E}^c}\) for a subset \(\mathcal{E}\subseteq A-B\). Such subsets are also witnesses for \(D(A)\leq D(B)\) because Conditions [D:degsub2b] and [D:degsub2c] of Definition 10 holds trivially true since there are no triangles entirely contained in \(D(A)\). Therefore, Part [T:pos-Degn1b] is proved.

Let us now prove Part [T:pos-Degn1a]. First of all, we have that \(W_{\delta}<\emptyset\) with \(W_{\delta}^{pr}\) and \(\widetilde{W_{\delta}^{pr}}\) being two witnesses since \(W_{\delta}=W_{\delta}^{pr}\sqcup \widetilde{W_{\delta}^{pr}}\) and any triangle contained in \(W_{\delta}\) contains always two elements of \(W_{\delta}^{pr}\) and one element of \(W_{\delta}^{pr}\), so that the conditions of Definition 10[D:degsub2] hold true.

It remains to show that \(W_{\delta}^{pr}\) and \(\widetilde{W_{\delta}^{pr}}\) are the only two witness for \(W_{\delta}<\emptyset\). So suppose that \(E\) is a witness for \(W_{\delta}<\emptyset\). Up to switching \(E\) with \(E^c\), we can assume that \((\delta+2;0)\in E\). Then we have to show that \(E=W_{\delta}^{pr}\). Using Condition [D:degsub2a] of Definition 10, it is enough to show that \(W_{\delta}^{pr}\subseteq E\). We will show this in three steps:

\(\star\) \(W_{2\delta}\subseteq E\).

Given \((e;h)\in W_{2\delta}\) we can write \((e;h)=(\delta+2;0)\circ_{h+1} (\delta+2;0)\) and then consider the triangle \(\Delta=[(\delta+2;0),(\delta+2;0),(e;h)^c)]\) in \(W_{\delta}\) (see 112 ): since the first two elements belong to \(E\), Condition [D:degsub2c] of Definition 10 implies that the last element does not belong to \(E\), which is equivalent to \((e;h)\in E\).

\(\star\) \(W_{\delta}\subseteq E\).

Given \((e;h)\in W_{\delta}\) we can form \((e;h)\circ_{e-1} (e;h)=(2;2h+e-1)\in W_{2\delta}\) and then consider the triangle \(\Delta=[(e;h),(e;h),(2;2h-2+e)^c)]\) in \(W_{\delta}\) (see 112 ): since the last element belongs to \(\widetilde{E}\) (and hence does not belong to \(E\)) and the first two elements are equal, Condition [D:degsub2c] of Definition 10 implies that \((e;h)\in E\).

\(\star\) \(W_{m\delta}\subseteq E\) for any \(m\geq 1\) such that \(m\delta\leq 2g-2\).

We prove this by induction on \(m\). The base case \(m=1\) has already been proved. Assume that \(m\geq 2\). Any element \((e;h)\in W_{m\delta}\) is such that \((e;h)> (\delta+2;0)\) since \(h\geq 0\) and \[2(e+h)=\delta(e;h)+2+e\geq 2\delta+4=2(\delta+2).\] Then, 115 implies that we can write \((e;h)=(\delta+2;0)\circ_i (\widetilde{e};\widetilde{h})\) for some \((\widetilde{e};\widetilde{h})\) which belongs to \(W_{(m-1)\delta}\) by 111 and hence to \(E\) by induction. Consider the triangle \(\Delta=[(\delta+2;0),(\widetilde{e};\widetilde{h}),(e;h)^c)]\) in \(W_{\delta}\) (see 112 ): since the first two elements belong to \(E\), Condition [D:degsub2c] of Definition 10 implies that the last element does not belong to \(E\), which is equivalent to \((e;h)\in E\).

It remains to prove that there are no further strict order relations between the elements of \(\operatorname{^{ns}\operatorname{Deg}}_{g,1}^{re}\). Using the classification of Propositions 38 and 39, we will divide the proof in three cases:

Case I: \(D(A)<D\in \operatorname{^{ns}\operatorname{Deg}}_{g,1}^{re}\Rightarrow D=D(B)\) for some \(A\supsetneq B\).

This follows from the fact that if \(D(A)<D\) then \(D(A)\supsetneq D\) together with the fact that:

\(\bullet\) \(D(A)\not\supseteq W_{\delta}\) for any \(1\leq \delta \leq g-1\);

\(\bullet\) \(D(A)\supsetneq D(B)\Leftrightarrow A\supsetneq B\).

Case II: \(W_{\delta}\not\leq W_{\widetilde{\delta}}\) for any \(1\leq \delta\neq \widetilde{\delta}\leq g-1\).

Indeed, this is trivially true if \(\widetilde{\delta}\) is not a multiple of \(\delta\), for otherwise \(W_{\delta}\not\supseteq W_{\widetilde{\delta}}\). Hence, we can assume that \(\widetilde{\delta}=k\delta\) for some \(k\geq 2\). Suppose by contradiction that \(W_{\delta}\leq W_{k \delta}\) and let \(E\subset W_{\delta}-W_{k\delta}\) a witness. Up to switching \(E\) with \(E^c\), we can assume that \((\delta+2;0)\in E\).

We will now prove by induction on \(m\) that \[\label{E:inclu-E} W_{m\delta}\subset E \text{ for any } 2\leq m \leq k-1.\tag{144}\] Indeed, for the base case \(m=2\) of the induction we argue as follows. Given \((e;h)\in W_{2\delta}\) we can write \((e;h)=(\delta+2;0)\circ_{h+1} (\delta+2;0)\) and then consider the triangle \(\Delta=[(\delta+2;0),(\delta+2;0),(e;h)^c)]\) (see 112 ): the first two elements belong to \(E\) and the last element belong to \(W_{2\delta}\subset W_{\delta}-W_{k\delta}\); hence Condition [D:degsub2c] of Definition 10 implies that the last element does not belong to \(E\), which is equivalent to \((e;h)\in E\). Now, for the induction step assume that \(3\leq m \leq k-1\). Any element \((e;h)\in W_{m\delta}\) is such that \((e;h)> (\delta+2;0)\) since \(h\geq 0\) and \[2(e+h)=\delta(e;h)+2+e>2\delta+4=2(\delta+2).\] Then, 115 implies that we can write \((e;h)=(\delta+2;0)\circ_i (\widetilde{e};\widetilde{h})\) for some \((\widetilde{e};\widetilde{h})\) which belongs to \(W_{(m-1)\delta}\) by 111 and hence to \(E\) by induction. Consider the triangle \(\Delta=[(\delta+2;0),(\widetilde{e};\widetilde{h}),(e;h)^c)]\) (see 112 ): the first two elements belong to \(E\) and the last element belong to \(W_{m\delta}\subset W_{\delta}-W_{k\delta}\); hence Condition [D:degsub2c] of Definition 10 implies that the last element does not belong to \(E\), which is equivalent to \((e;h)\in E\). The proof of 144 is now complete.

We now finish the proof by finding the desired contradiction. Since \((\delta+2;0)\circ_i ((k-1)\delta+2;0)=(k\delta+2;0)\), we can look at the triangle \(\Delta=[(\delta+2;0),((k-1)\delta+2;0),(k\delta+2;0)^c]\) of \(W_{\delta}\) (see 112 ): the first two elements of \(\Delta\) belong to \(E\) by the assumption that \((\delta+2;0)\in E\) and 144 , while the last element belongs to \(W_{k\delta}\), and this violates Condition [D:degsub2b] of Definition 10.

Case III: \(W_{\delta}\not\leq D(A)\) for any non-empty antichain \(A\) of \(\mathbb{D}_{g,1}^{\geq g}\).

Indeed, first of all, using [T:pos-Degn1b], we can assume that \(A=\{x\}\) for some \(x\in \mathbb{D}_{g,1}^{\geq g}\), so that \(D(A)=\{x,x^c\}\). Moreover, the statement is trivially true if \(\delta(x)\) is not a multiple of \(\delta\), for otherwise \(W_{\delta}\not\supseteq \{x, x^c\}\). Hence, we can assume that \(\delta(x)=k\delta\) for some \(k\geq 2\) (since \(\delta(x)\geq g\) while \(\delta\leq g-1\)). Suppose by contradiction that \(W_{\delta}\leq \{x\}\) and let \(E\subset W_{\delta}-\{x,x^c\}\) be a witness. Up to switching \(E\) and \(E^c\), we can assume that \((\delta+2;0)\in E\).

Observe that 144 holds also in this case with the same proof of the Case II.

We can now find the desired contradiction. Write \(x=(e;h)\) and observe that \((e;h)>(\delta+2;0)\) since \(h\geq 0\) and \[2(h+e)=\delta(e;h)+e+2=k\delta+e+2\geq 2\delta+4=2(\delta+2).\] Then, 115 implies that we can write \((e;h)=(\delta+2;0)\circ_i (\widetilde{e};\widetilde{h})\) for some \((\widetilde{e};\widetilde{h})\) which belongs to \(W_{(k-1)\delta}\) by 111 and hence to \(E\) by 144 . Consider the triangle \(\Delta=[(\delta+2;0),(\widetilde{e};\widetilde{h}),(e;h)^c=x^c)]\) (see 112 ): the first two elements belong to \(E\) and the last element belongs to \(\{x,x^c\}\), and this violates Condition [D:degsub2b] of Definition 10. ◻

Proof. By Lemma 3 and Proposition 34, it is enough to show that \(\operatorname{^{ns}\Sigma}_{g,1}^{\chi}\) is connected through height one. This follows from Lemmas 11 and 12 below, using that the relation of being connected through height one is an equivalence relation on the set of maximal elements of \(\operatorname{^{ns}\Sigma}_{g,1}^\chi\). ◻

Proof. Since the poset of (non-separating) classical V-functions is the poset of regions of an hyperplane arrangement in a real affine space by Lemma 1, then is connected through height \(1\). Thus it is enough to show that any two non-separating classical V-functions \(\sigma\) and \(\sigma'\), which are adjacent in the poset of regions, are connected through height \(1\) in the poset \(\operatorname{^{ns}\Sigma}^\chi_{g,1}\).

Let \(\overline{\sigma}\in\operatorname{^{ns}\Sigma}_{g,1}^\chi\) be a classical V-function such that \(\sigma,\sigma'>\overline{\sigma}\) and let \(D=\mathcal{D}(\overline{\sigma})\). If \(D\) is a submaximal element of \(\operatorname{Deg}_{g,1}\), we are done. Otherwise, \(D=W_{\delta}\), for some \(\delta\) such that \(g\leq\delta\leq 2g-4\) by Theorem 37[T:class-n1i] and Corollary 8. In particular, \(|D|=2k\geq 2\).

We observe that, since \(\sigma>\overline{\sigma}<\sigma'\) and \(\mathcal{D}(\overline{\sigma})=D\), we have that \(\sigma_{|_{D^c}}=\sigma'_{|_{D^c}}\). Let \(\{(e_i;h_i)\}_{i=1,\ldots,k}\) be such that \(D=\bigcup_{i=1}^k\{(e_i;h_i),(e_i;h_i)^c\}\), and let \(D_i:=\{(e_i;h_i),(e_i;h_i)^c\}\) for each \(i=1\ldots,k\). Since both \(\sigma\) and \(\sigma'\) are maximal non-separating V-functions, for any \(i=1,\ldots,k\), the function \(\sigma_i\) defined as \[\sigma_i(e;h,A):=\begin{cases} \sigma'(e;h,A),\mathrm{ if }(e;h,A)\in\bigcup_{j\leq i}D_j;\\ \sigma(e;h,A), \mathrm{ otherwise}, \end{cases}\] is also a maximal non-separating V-function. In particular, \(\sigma=\sigma_0\) (trivially) and \(\sigma'=\sigma_k\).

Similarly, for each \(i=1,\ldots,k-1\), the function \(\overline{\sigma}_i\), defined as \[\overline{\sigma}_i(e;h,A):=\min (\sigma_i(e;h,A),\sigma_{i+1}(e;h,A)) \quad \text{for all } (e;h,A) \in {^{ns}\mathbb{D}_{g,1}}\] is a non-separating V-function such that \(\mathcal{D}(\overline{\sigma}_i)=D_i\) (in particular \(\overline{\sigma}_i\) is submaximal) and \(\sigma_i>\overline{\sigma}_i<\sigma_{i+1}\).

Therefore, we conclude that \(\sigma\) and \(\sigma'\) are connected through height \(1\) via the sequence
\(\sigma=\sigma_0,\overline{\sigma}_0,\sigma_1,\ldots,\overline{\sigma}_{k-1},\sigma_k=\sigma'\). ◻

Proof. If \(\sigma\) is classical there is nothing to prove. Otherwise, by Theorem 37, there exists \(\delta'\) such that \(\sigma\) is not constant on \(\mathbb{D}_{\delta'}:=\{x\in \mathbb{D}_{g,1}^{pr}: \delta(x)=\delta'\}\). Let \(\delta\) be the minimum such degree \(\delta'\). We want to show that \(\sigma\) is connected through height \(1\) to a non-separating V-function that is constant on \(\mathbb{D}_{\delta}\).

Since \(\sigma\) is a V-function, by 120 and minimality of \(\delta\), for each \((e;h),(e';h')\in \mathbb{D}_{\delta}\), we have \(|\sigma(e;h)-\sigma(e';h')|\leq~1\). Thus, we can write \(\mathbb{D}_{\delta}=D_1\sqcup D_2\) with \(D_1,D_2\neq \emptyset\) such that \[\label{E:sigmaD1D2} \sigma(e;h)=\sigma(e';h')-1\quad\mathrm{for all } (e;h)\in D_1,(e';h')\in D_2.\tag{145}\] We observe, in particular, that \(\sigma\) is constant on \(D_1\) and \(D_2\).

Let \((e_0;h_0)\in D_1\) with \((e_0;h_0)\) minimal with respect to the order in 113 .

The function \(\overline{\sigma}\) defined by \[\overline{\sigma}(e;h,A):=\begin{cases} \sigma(e;h,A)-1,\mathrm{ if }(e;h,A)=(e_0;h_0)^c;\\ \sigma(e;h,A), \mathrm{ otherwise}, \end{cases}\] is a V-function of type \((g,1)\) such that \(\mathcal{D}(\overline{\sigma})=\{(e_0;h_0),(e_0;h_0)^c\}\) and \(\overline{\sigma}<\sigma\).

Indeed, since \(\sigma\) is a V-function with \(\mathcal{D}(\sigma)=\emptyset\) and since there are no triangles contained in \(\{(e_0;h_0),(e_0;h_0)^c\}\), by 14 , we only need to check that, for each triangle \(\Delta=[(e_0;h_0)^c,(e_1;h_1),(e_2;h_2)]\), we have \[\overline{\sigma}((e_0;h_0)^c)+\overline{\sigma}(e_1;h_1)+\overline{\sigma}(e_2;h_2)-\chi=1,\] that is \[\sigma((e_0;h_0)^c)+\sigma(e_1;h_1)+\sigma(e_2;h_2)-\chi=2.\] Suppose, by contradiction, that there exists \(\Delta\) such that \(\sigma((e_0;h_0)^c)+\sigma(e_1;h_1)+\sigma(e_2;h_2)-\chi=1\). By the assumptions of minimality on \(\delta\) and \((e_0;h_0)\), \(\sigma\) is constant on \(\mathbb{D}_{\delta_i}\), where \(\delta_i:=\delta(e_i;h_i)\), for \(i=1,2.\) Clearly, we can choose \((e'_1;h'_1),(e'_2;h'_2)\) with \(\delta(e'_i;h'_i)=\delta_i\), such that \(\Delta=[(e;h)^c,(e'_1;h'_1),(e'_2;h'_2)]\) is a triangle with \((e;h)\in D_2\). So, by 145 , we have \[\sigma((e;h)^c)+\sigma(e'_1;h'_1)+\sigma(e'_2;h'_2)-\chi=0,\] a contradiction, which finishes the proof of the Claim.

Consider now the function \(\sigma'\) defined as \[\sigma'(e;h,A):=\begin{cases} \overline{\sigma}(e;h,A)+1,\mathrm{ if }(e;h,A)=(e_0;h_0);\\ \overline{\sigma}(e;h,A), \mathrm{ otherwise}, \end{cases}\] which is a V-function such that \(\sigma'>\overline{\sigma}<\sigma\), hence it is connected to \(\sigma\) through height \(1\). Moreover \(\sigma'\) satisfies: \[\sigma'(e;h)=\sigma'(e';,h')-1\quad\mathrm{for all } (e;h)\in D_1-\{(e_0;h_0)\},\mathrm{ and }(e';,h')\in D_2\cup\{(e_0;h_0)\}.\]

By inductively repeating the same argument for each element of \(D_1\), we obtain a V-function that is constant on \(\mathbb{D}_{\delta}\) and is connected to \(\sigma\) through height \(1\). Finally, repeating again the same argument for each \(\mathbb{D}_{\delta'}\) such that \(\sigma\) is not constant on \(\mathbb{D}_{\delta'}\), we obtain a V-function \(\tau\) that is connected through height \(1\) to \(\sigma\) and that is uniform, hence classical by Theorem 37. ◻

Proof. Let \(D\in\operatorname{Deg}_{g,n}\). We have to prove that \(\emptyset\ge D\) in \(\operatorname{Deg}_{g,n}\). Since \(\emptyset\subseteq D\), it remains to exhibit a witness for the relation \(\emptyset \geq D\), i.e. a subset \(E\subseteq D\) satisfying the conditions of Definition 10[D:degsub2].

Fix an index \(j\in[n]\) and set \[E \;:=\; \{\, (e;h,A)\in D \;:\; j\in A \,\}.\] We check the required conditions.

[D:degsub2a]: If \((e;h,A)\in D\), then its complement is \((e;g-h-e+1,A^c)\in D\) (since \(D\) is a degeneracy subset). Exactly one of \(A\) and \(A^c\) contains \(j\), hence \((e;h,A)\in E\) if and only if \((e;g-h-e+1,A^c)\notin E\), i.e.the complement of \((e;h,A)\) lies in \(E^c\).

[D:degsub2b]: This condition is vacuous because it concerns triangles \(\Delta\subseteq D\) satisfying \(|\Delta\cap D_1|=1\), and \(D_1=\emptyset\).

[D:degsub2c]: Let \(\Delta=[(e_i;h_i,A_i)]_{i=1}^3\) be a triangle contained in \(D\). By definition of triangle we have a disjoint union \(A_1\sqcup A_2\sqcup A_3=[n]\), so exactly one of the \(A_i\) contains \(j\). Therefore \(|\Delta\cap E|=1\), and (2c) is satisfied. ◻

Proof. This follows from Proposition 44 and Remark 43, together with the fact that \(\sigma\in \Sigma_{g,n}\) is general if and only if \(\mathcal{D}(\sigma)=\emptyset\). ◻

Proof. We can assume that \(n\geq 2\), for otherwise the statement is a tautology.

Let us first prove Part [P:reduceto1a]. Since the elements of \(\operatorname{Deg}_{g,n}^u\) are exactly the elements \(D'\in \operatorname{Deg}_{g,n}\) such that \(D'\subseteq \mathbb{D}_{g,n}^u=\operatorname{Im}(\alpha_n)\), the fact that \(\alpha_{n*}\) is a well-defined bijection is equivalent to showing that a subset \(D\subset \mathbb{D}_{g,1}\) belongs to \(\operatorname{Deg}_{g,1}\), i.e. it satisfies the two conditions of Definition 10[D:degsub1], if and only if \(\alpha_n(D)\) belongs to \(\operatorname{Deg}_{g,n}\). The complement-closure for \(D\) is equivalent to the complement-closure for \(\alpha_{n*}(D)\) since \(\alpha_n(x^c)=\alpha_n(x)^c\) for any \(x\in \mathbb{D}_{g,1}\). The triangle-closure for \(\alpha_{n*}(D)\) implies the triangle-closure for \(D\) since \(\alpha_n\) induces a bijection between the triangles of \(\mathbb{D}_{g,1}\) and the triangles of \(\mathbb{D}_{g,n}\) entirely contained in \(\operatorname{Im}(\alpha_n)\). On the other hand, the triangle-closure for \(D\) implies the triangle-closure for \(\alpha_n(D)\) using that if a triangle of \(\mathbb{D}_{g,n}\) is such that two of its elements belong to \(\operatorname{Im}(\alpha_n)\) then also the third one belongs to \(\operatorname{Im}(\alpha_n)\) and hence it is the image of a triangle of \(\mathbb{D}_{g,1}\) via \(\alpha_n\).

The partial order on each \(\operatorname{Deg}_{g,k}\) (Definition 10[D:degsub2] is defined using inclusions and the existence of a “witness” subset \(E\), via conditions that are expressed only in terms of complements and triangles. If \(D_1\ge D_2\) in \(\operatorname{Deg}_{g,1}\) with witness \(E\subseteq D_2- D_1\), then \(\alpha_n(E)\subseteq \alpha_n(D_2)- \alpha_n(D_1)\) satisfies the corresponding witness conditions for \(\alpha_n(D_1)\ge \alpha_n(D_2)\) because \(\alpha_n\) preserves complements and identifies triangles as above. Conversely, any witness for \(\alpha_n(D_1)\ge \alpha_n(D_2)\) pulls back via \(\alpha_n\) to a witness for \(D_1\ge D_2\). Thus \(\alpha_{n*}\) is an order-isomorphism onto its image.

Let us now prove Part [P:reduceto1b] considering the two implications separately.

(\(\Leftarrow\)) Let \(D\in \operatorname{Deg}_{g,1}\) be such that \(\alpha_{n*}(D)\) is realizable, i.e. \(\alpha_n(D)=\mathcal{D}(\sigma)\) for some \(\sigma\in \Sigma_{g,n}^\chi\). The composition \[\alpha_n^*(\sigma):\mathbb{D}_{g,1}\xrightarrow{\alpha_n} \mathbb{D}_{g,n}\xrightarrow{\sigma} \mathbb{Z}\] defines an element of \(\Sigma_{g,1}^\chi\) (using that \(\alpha_n\) commutes with taking complements and it sends triangles of \(\mathbb{D}_{g,1}\) into triangles of \(\mathbb{D}_{g,n}\)) such that \[\mathcal{D}(\alpha_n^*(\sigma))=\alpha_n^{-1}(\mathcal{D}(\sigma))=\alpha_n^{-1}(\alpha_n(D))=D,\] which shows that \(D\) is realizable.

(\(\Rightarrow\)) Let \(D\in \operatorname{Deg}_{g,1}\) be realizable, i.e. \(D=\mathcal{D}(\sigma)\) for some \(\sigma\in \Sigma_{g,1}^\chi\). In order to show that \(\alpha_{n*}(D)\) is realizable, we partition the set \(\mathbb{D}_{g,n}\) into four disjoint subsets \[\mathbb{D}_{g,n}=\mathbb{D}_{g,n}^u\sqcup \mathbb{D}_{g,n}^i \sqcup \mathbb{D}_{g,n}^< \sqcup \mathbb{D}_{g,n}^>,\] where \(\mathbb{D}_{g,n}^u\) is the set of unmixed elements (see Definition 13), \(\mathbb{D}_{g,n}^i\) is the set of intermediate elements: \[\mathbb{D}_{g,n}^i:=\{(e;h,A)\in \mathbb{D}_{g,n}: \emptyset\subsetneq A \subsetneq [n] \text{ and } 0<2h-2+e+|A-\{2,\ldots, n\}|<2g-1\},\] \(\mathbb{D}_{g,n}^<\) is the set of small elements \[\begin{align} &\mathbb{D}_{g,n}^<:=\{(e;h,A)\in \mathbb{D}_{g,n}: 2h-2+e+|A-\{2,\ldots, n\}|\leq 0\}= \\ & =\{(e;h,A)\in \mathbb{D}_{g,n}:\text{ either } (e,h)=(1,0) \text{ or } (e,h)=(2,0) \text{ and } 1\not \in A\}, \end{align}\] \(\mathbb{D}_{g,n}^>\) is the set of big elements \[\begin{align} &\mathbb{D}_{g,n}^>:=\{(e;h,A)\in \mathbb{D}_{g,n}: 2h-2+e+|A-\{2,\ldots, n\}|\geq 2g-1\}= \\ & =\{(e;h,A)\in \mathbb{D}_{g,n}:\text{ either } (e,h)=(1,g) \text{ or } (e,h)=(2,g-1) \text{ and } 1 \in A\}, \end{align}\] Note that there exists a forgetful map \[\begin{align} \varpi: \mathbb{D}_{g,n}^u\sqcup \mathbb{D}_{g,n}^i & \longrightarrow \mathbb{D}_{g,1}\\ (e;h,A)& \mapsto (e;h,A-\{2,\ldots, n\}) \end{align}\] such that \(\varpi\circ \alpha_n=\operatorname{id}\).

Consider the following function \[\begin{align} \widetilde{\sigma}: \mathbb{D}_{g,n}& \longrightarrow \mathbb{Z}\\ (e;h,A) & \mapsto { \left\{\begin{aligned} & 0 & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^< \text{ and } n\not\in A,\\ & 1 & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^< \text{ and } n\in A, \\ & \chi & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^> \text{ and } n\not\in A, \\ & \chi+1 & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^> \text{ and } n\in A, \\ & \sigma(\varpi(e;h,A)) & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^u,\\ & \sigma(\varpi(e;h,A)) & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^i \text{ and either } \varpi(e;h,A)\not \in \mathcal{D}(\sigma) \text{ or } n\not \in A,\\ & \sigma(\varpi(e;h,A))+1 & \text{ if } (e;h,A)\in \mathbb{D}_{g,n}^i \text{ and } \varpi(e;h,A) \in \mathcal{D}(\sigma) \text{ and } n\in A.\\ \end{aligned}\right.} \end{align}\] We claim that \[\widetilde{\sigma}\in \Sigma_{g,n}^\chi \text{ with } \mathcal{D}(\widetilde{\sigma})=\alpha_{n*}(D),\] which will conclude the proof.

We need to check the two conditions of Definition 5 for \(\widetilde{\sigma}\).

\(\bullet\) Condition [E:condUni1]: since the complementary involution on \(\mathbb{D}_{g,n}\) maps \(\mathbb{D}_{g,n}^>\) isomorphically onto \(\mathbb{D}_{g,n}^<\) and it preserves \(\mathbb{D}_{g,n}^u\) and \(\mathbb{D}_{g,n}^i\), and since the forgetful map commutes with the complementary involutions on the source and target, we deduce that \(\widetilde{\sigma}\) satisfies Equation 13 with \[\mathcal{D}(\widetilde{\sigma})=\varpi^{-1}(\mathcal{D}(\sigma))\cap \mathbb{D}_{g,n}^u=\alpha_{n*}(\mathcal{D}(\sigma))=\alpha_{n*}(D).\]

\(\bullet\) Condition [E:condUni2]: the first Condition (2a) holds true since we already know that \(\mathcal{D}(\widetilde{\sigma})=\alpha_{n*}(D)\in \operatorname{Deg}_{g,n}\). In order to check Condition (2b), consider a triangle \(\Delta=[x_1,x_2,x_3]\) of \(\mathbb{D}_{g,n}\), with \(x_i:=(e_i;h_i,A_i)\) for \(i=1,2,3\), and consider the following different possibilities:

(a) Assume that \(\Delta\) contains some elements of \(\mathbb{D}_{g,n}^>\sqcup \mathbb{D}_{g,n}^<\). Then there are two possibilities:

(a1) If \(\Delta\) contains at least two elements of \(\mathbb{D}_{g,n}^>\sqcup \mathbb{D}_{g,n}^<\), then we must have (up to reordering) that \(x_1=(2;0,A_1),x_2=(2;0,A_2)\in \mathbb{D}_{g,n}^<\) and \(x_3=(2;g-1,A_3)\in \mathbb{D}_{g,n}^>\), and \(n\) must belong to exactly one among \(A_1, A_2, A_3\), so that \[\widetilde{\sigma}(x_1)+\widetilde{\sigma}(x_2)+\widetilde{\sigma}(x_3)=0+0+\chi+1=\chi+1,\] and 14 is satisfied since all the elements of \(\Delta\) are \(\widetilde{\sigma}\)-nondegenerate.

(a2) If \(\Delta\) contains exactly one element of \(\mathbb{D}_{g,n}^>\sqcup \mathbb{D}_{g,n}^<\), then we must have (up to reordering) that \(x_1=(2;0,A_1)\in \mathbb{D}_{g,n}^<\) and \(x_2,x_3\in \mathbb{D}_{g,n}^u\sqcup \mathbb{D}_{g,n}^i\) with \(\varpi(x_2)^c=\varpi(x_3)\). Moreover, \(A_1\) must be non-empty and it cannot contain \(1\), so that if we assume that \(1\in A_2\) (up to reordering) then we have \(x_2\in \mathbb{D}_{g,n}^i\), while \(x_3\in \mathbb{D}_{g,n}^u\) if and only if \(A_3=\emptyset\). We now compute \[\widetilde{\sigma}(x_1)+\widetilde{\sigma}(x_2)+\widetilde{\sigma}(x_3)= \begin{cases} 0/1+\chi+1 & \text{ if }\varpi(x_2), \varpi(x_3)\not \in \mathcal{D}(\sigma),\\ \chi+1 & \text{ if }\varpi(x_2), \varpi(x_3)\in \mathcal{D}(\sigma) \text{ and } x_3\in \mathbb{D}_{g,n}^i,\\ \chi+1 & \text{ if }\varpi(x_2), \varpi(x_3)\in \mathcal{D}(\sigma) \text{ and } x_3\in \mathbb{D}_{g,n}^u,\\ \end{cases}\] and 14 is satisfied since in the first two cases all the elements of \(\Delta\) are \(\widetilde{\sigma}\)-nondegenerate, while in the third case only \(x_3\in \mathcal{D}(\widetilde{\sigma})\).

(b) Assume that \(\Delta\) does not contain elements of \((\mathbb{D}_{g,n}^>\sqcup \mathbb{D}_{g,n}^<)\) but it does contain some element of \(\mathbb{D}_{g,n}^u\). We are going to use that \(\sigma\) verifies 14 for the triangle \(\varpi(\Delta)=[\varpi(x_1),\varpi(x_2),\varpi(x_3)]\) of \(\mathbb{D}_{g,1}\). There are two possibilities:

(b1) If \(\Delta\) contains at least two elements of \(\mathbb{D}_{g,n}^u\), then \(\Delta\) is entirely contained in \(\mathbb{D}_{g,n}^u\) and we have that \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)=\sum_{i=1}^3\sigma(\varpi(x_i))\] and hence 14 for \(\widetilde{\sigma}\) follows from 14 for \(\sigma\).

(b2) If \(\Delta\) contains exactly one element of \(\mathbb{D}_{g,n}^u\), say \(x_1\), then we must have that \(x_2,x_3\in \mathbb{D}_{g,n}^i\) (which implies that \(x_2,x_3\not \in \mathcal{D}(\widetilde{\sigma})\)) and that \(A_1=\emptyset\) (which implies that \(n\in A_2\sqcup A_3\)). Now there are four cases:

\(\star\) If \(\varpi(x_1)\in \mathcal{D}(\sigma)\) (which implies that \(x_1\in \mathcal{D}(\widetilde{\sigma})\)) and \(\varpi(x_2),\varpi(x_3)\not \in \mathcal{D}(\sigma)\) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi=1,\] and hence 14 holds true.

\(\star\) If \(\varpi(x_1)\in \mathcal{D}(\sigma)\) (which implies that \(x_1\in \mathcal{D}(\widetilde{\sigma})\)) and one among \(\varpi(x_2)\) and \(\varpi(x_3)\) belongs to \(\mathcal{D}(\sigma)\) (which then implies that both of them belong to \(\mathcal{D}(\sigma)\)) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi+1=1,\] and hence 14 holds true.

\(\star\) If \(\varpi(x_1)\not \in \mathcal{D}(\sigma)\) (which implies that \(x_1\not \in \mathcal{D}(\widetilde{\sigma})\)) and \(\varpi(x_2),\varpi(x_3)\not \in \mathcal{D}(\sigma)\) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi\in \{1,2\},\] and hence 14 holds true.

\(\star\) If \(\varpi(x_1)\not \in \mathcal{D}(\sigma)\) (which implies that \(x_1\not \in \mathcal{D}(\widetilde{\sigma})\)) and one among \(\varpi(x_2)\) and \(\varpi(x_3)\) belongs to \(\mathcal{D}(\sigma)\) (which then implies that exactly one of them belong to \(\mathcal{D}(\sigma)\)) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi+0/1=1+0/1\in \{1,2\},\] and hence 14 holds true.

(c) Assume that \(\Delta\) is entirely contained in \(\mathbb{D}_{g,n}^i\), so that none of the elements of \(\Delta\) belongs to \(\mathcal{D}(\widetilde{\sigma})\). We are going to use that \(\sigma\) verifies 14 for the triangle \(\varpi(\Delta)=[\varpi(x_1),\varpi(x_2),\varpi(x_3)]\) of \(\mathbb{D}_{g,1}\). There are three cases:

\(\star\) If \(\varpi(x_1),\varpi(x_2),\varpi(x_3)\in \mathcal{D}(\sigma)\) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi+1=1,\] and hence 14 holds true.

\(\star\) If only one among \(\{\varpi(x_1),\varpi(x_2),\varpi(x_3)\}\) belongs to \(\mathcal{D}(\sigma)\) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi+0/1=1+0/1\in \{1,2\},\] and hence 14 holds true.

\(\star\) If none of the elements of \(\{\varpi(x_1),\varpi(x_2),\varpi(x_3)\}\) belongs to \(\mathcal{D}(\sigma)\) then \[\sum_{i=1}^3 \widetilde{\sigma}(x_i)-\chi=\sum_{i=1}^3\sigma(\varpi(x_i))-\chi\in \{1,2\},\] and hence 14 holds true. ◻

Proof. This follows from the previous Proposition together with the fact that the \(\operatorname{Deg}^u_{g,n}\) is upward-closed in \(\operatorname{Deg}_{g,n}\). ◻

Proof. We divide the proof in two steps.

Step I: \(\{x,x^c\}\) is a (mixed) submaximal element for any \(x=(e;h,A)\in \mathbb{D}_{g,n}\) with \(\emptyset \subsetneq A \subsetneq [n]\).

Indeed, let us first show that \(\{x,x^c\}\in \operatorname{Deg}_{g,n}\), i.e. that \(\{x,x^c\}\) satisfies the two conditions of Definition 10[D:degsub1]. Complement-closure (i.e. Condition 10[D:degsub1a]) is clear. For the triangle-closure (i.e. Condition 10[D:degsub1b]), we show that no triangle in \(\mathbb{D}_{g,n}\) contains two elements of \(\{x,x^c\}\). The only way for two elements of \(\{x,x^c\}\) to appear in a triangle \(\Delta\) is:

1. \(\Delta=[x,x,y]\) for some \(y\in \mathbb{D}_{g,n}\), but this is absurd since \(A\neq \emptyset\);

2. \(\Delta=[x^c,x^c,y]\) for some \(y\in \mathbb{D}_{g,n}\), but this is absurd since \(A^c\neq \emptyset\);

3. \(\Delta=[x,x^c,y]\) for some \(y\in \mathbb{D}_{g,n}\), but this absurd since 11 would imply that \[\delta(y)=2g-2+n-\delta(x)-\delta(x^c)=0,\] which is impossible since \(\delta(y)\in (0,2g-2+n)\).

Now it is clear that \(\{x,x^c\}\) is a submaximal element of \(\operatorname{Deg}_{g,n}\) since it is dominated only by \(\emptyset\).

Step II: If \(D\) is a (mixed) submaximal element of \(\operatorname{Deg}_{g,n}\) then \(D=\{x,x^c\}\) for some \(x=(e;h,A)\in \mathbb{D}_{g,n}\) with \(\emptyset \subsetneq A \subsetneq [n]\).

Choose \(x_0=(e_0;h_0,A_0)\in D\) with \(\emptyset\subsetneq A_0\subsetneq [n]\) by minimizing lexicographically first \(|A|\), then \(h\), then \(e\) among all \((e;h,A)\) occurring in \(D\). By Definition 10[D:degsub1a], \(D\) is closed under complements, hence \(x_0^c\in D\). Set \(D_0:=\{x_0,x_0^c\}\). By Step I, we have that \(D_0 \in \operatorname{Deg}_{g,n}\).

We will show that \(D \leq D_0\), which then will force the equality \(D=D_0\) (and hence the conclusion) because \(D\) is a submaximal element and \(D_0\) is not a maximal element by Proposition 44.

In order to show that \(D_0\leq D\), using that \(D_0 \subseteq D\), it suffices to find a witness \(E \subseteq D - D_0\). Since \(A_0\subsetneq[n]\), there exists \(j\in A_0^c\). We define \[E\;:=\;\{(e;h,A)\in D- D_0 \;:\;j\in A\}.\] We will check that \(E\) satisfies the conditions of Definition 10[D:degsub2].

[D:degsub2a]. Let \((e;h,A)\in D- D_0\). Then its complement is \((e;h,A)^c=(e;g-h-e+1,A^c)\in D- D_0\). Exactly one of \(A\) and \(A^c\) contains \(j\), so exactly one element in the complementary pair lies in \(E\), which settles Condition [D:degsub2a].

[D:degsub2b]. Let \(\Delta \subseteq D\) be a triangle such that \(|\Delta\cap D_0|=1\). We will show that \(|\Delta\cap E|=1\). There are two cases.

Case 1: \(\Delta\cap D_0=\{x_0^c\}\). We will show that this case cannot occur. Write \[x_0=(e_0;h_0,A_0)\in D_0,\qquad x_0^c=(e_0;g-h_0-e_0+1,A_0^c)\in D_0,\] and let the other two elements of \(\Delta\) be \[x_2=(e_2;h_2,A_2) \in D - D_0,\qquad x_3=(e_3;h_3,A_3)\in D- D_0.\] Since \(\Delta\) is a triangle, the marking sets satisfy \[A_0^c\sqcup A_2\sqcup A_3=[n]\qquad\text{hence}\qquad A_2\sqcup A_3=A_0.\] If both \(A_2\) and \(A_3\) are nonempty, then one of them is a proper nontrivial subset of \(A_0\), contradicting the minimality of \(|A_0|\). Thus it suffices to rule out the subcase \[(A_2,A_3)=(A_0,\emptyset)\quad\text{or}\quad(\emptyset,A_0).\] Without loss of generality, assume \((A_2,A_3)=(A_0,\emptyset)\) for a contradiction.

The triangle genus equation gives \[g=(g-h_0-e_0+1)+h_2+h_3+\frac{e_0+e_2+e_3}{2}-2,\] equivalently \[\label{eq:case1} h_2+h_3+\frac{e_2+e_3-e_0}{2}=h_0+1.\tag{146}\] Set \(\displaystyle n_{23}:=\frac{e_2+e_3-e_0}{2}\). The edge inequalities for a triangle imply \(n_{23}\ge 1\), so 146 yields \(h_2=h_0+1-h_3-n_{23}\le h_0\). By the defining choice of \(x_0\) (minimality of the marking set \(|A_0|\) and then of \(h_0\)), we must have \(h_2\ge h_0\), hence \(h_2=h_0\) and then \(h_3+n_{23}=1\). Therefore \(h_3=0\) and \(n_{23}=1\), i.e.\(e_2+e_3=e_0+2\). Since \(x_3\in \mathbb{D}_{g,n}\) and \(x_3\) belongs to a triangle, one has \(e_3\ge 2\), so \(e_2\le e_0\); by the minimality assumption and the fact that \(h_2=h_0\), we deduce \(e_2\ge e_0\), hence \(e_2=e_0\) and \(e_3=2\). Thus \(x_3=(2,0,\emptyset)\), contradicting the stability inequality \(2h-2+e+|A|>0\) for elements of \(\mathbb{D}_{g,n}\), so Case 1 cannot occur.

Case 2: \(\Delta\cap D_0=\{x_0\}\). Then one element of \(\Delta\) has marking set \(A_0\), so the other two marking sets partition \(A_0^c=A_i\sqcup A_k\). Since \(j\in A_0^c\), exactly one of \(A_i\) and \(A_k\) contains \(j\). By definition of \(E\), exactly one of the two elements in \(\Delta-\{x_0\}\) lies in \(E\). Also \(x_0\notin E\) because \(E\subseteq D- D_0\), therefore \(|\Delta\cap E|=1\).

This completes the proof of [D:degsub2b].

[D:degsub2c]. Let \(\Delta\subset D- D_0\) be a triangle. Then \(A_1\sqcup A_2\sqcup A_3=[n]\), so exactly one of the marking sets contains \(j\). Hence exactly one element of \(\Delta\) lies in \(E\), which means \(|\Delta\cap E|=1\). Condition [D:degsub2c] is satisfied. ◻

Proof. By Proposition 46, \(W=\{x,x^c\}\) for some \(x=(e_0;h_0,A_0)\in D_{g,n}\) with \(\emptyset\subsetneq A_0\subsetneq [n]\).

Set \(\chi:=0\), choose \(j\not \in A_0\) and define \(\sigma:\mathbb{D}_{g,n}\to\mathbb{Z}\) as follows:

• \(\sigma(x)=1\) and \(\sigma(x^c)=-1\).

• If \(y=(e;h,A)\) satisfies \(j\notin A\) and \(y\neq x\), set \[\sigma(y):=\left\lceil\frac{\delta(y)}{\delta(x)}\right\rceil \quad \text{ and } \quad \sigma(y^c):=1-\sigma(y).\]

We now check that \(\sigma\in \Sigma_{g,n}^0\) with \(\mathcal{D}(\sigma)=\{x,x^c\}\), by verifying the conditions of Definition 5.

Condition [E:condUni1] is clearly satisfied with \(\chi=0\) and the \(\sigma\)-degenerate elements are \(\{x,x^c\}\).

Let us now check Condition [E:condUni2]. First of all, Condition (2a) holds vacuously because no triangle can contain two elements of \(\{x,x^c\}\) as we already observed in Step I of the proof of Proposition 46. Thus, it remains to verify Condition (2b). Let \(\Delta=[u,v,w^c]\) be any triangle in \(\mathbb{D}_{g,n}\), arranged so that \(j\) belong to the marked set of \(w^c\) (and hence it does not belong to the marked set of \(u\), \(v\) and \(w\)). By 11 , we have \(\delta(w)=\delta(u)+\delta(v)\). We check 14 (with \(\chi=0\)) for \(\Delta\) by distinguishing the following three exhausting cases.

Case 1: \(\Delta\cap\{x,x^c\}=\emptyset\). Using \(\delta(w)=\delta(u)+\delta(v)\) and the definition of \(\sigma\), we get \[\sigma(u)+\sigma(v)-\sigma(w) =\left\lceil \frac{\delta(u)}{\delta(x)}\right\rceil +\left\lceil \frac{\delta(v)}{\delta(x)}\right\rceil -\left\lceil \frac{\delta(u)+\delta(v)}{\delta(x)}\right\rceil\in\{0,1\}.\] From this and the fact that \(\sigma(w^c)=1-\sigma(w)\), we compute \[\sigma(u)+\sigma(v)+\sigma(w^c) = 1+\sigma(u)+\sigma(v)-\sigma(w)\in \{1,2\},\] which verifies Equation 14 since the three elements of \(\Delta\) are all \(\sigma\)-nondegenerate.

Case 2: \(w^c=x^c\) (so \(w=x\)). Using that \(\delta(x)=\delta(w)=\delta(u)+\delta(v)\) and that \(\delta(u),\delta(v)>0\), we get that \(\sigma(u)=\sigma(v)=1\). Hence, using also \(\sigma(w^c)=\sigma(x^c)=-1\), we compute \[\sigma(u)+\sigma(v)+\sigma(w^c)=1+1-1=1,\] which verifies Equation 14 since \(x^c\) is the unique element of \(\Delta\) that is \(\sigma\)-degenerate.

Case 3: \(u=x\) or \(v=x\) (say \(u=x\)) and \(w\neq x\). Since \(\delta(w)=\delta(u)+\delta(v)=\delta(x)+\delta(v)\), we have \[\sigma(w)=\left\lceil\frac{\delta(x)+\delta(v)}{\delta(x)}\right\rceil =1+\left\lceil\frac{\delta(v)}{\delta(x)}\right\rceil =1+\sigma(v).\] Using this and the fact that \(\sigma(w^c)=1-\sigma(w)\) (because \(w^c\neq x^c\)), we compute \[\sigma(x)+\sigma(v)+\sigma(w^c)=1+\sigma(v)+1-\sigma(w)=1,\] which verifies Equation 14 since \(x\) is the unique element of \(\Delta\) that is \(\sigma\)-degenerate. ◻

Proof. By Remark 43, an element \(\sigma\in \Sigma_{g,n}^\chi\) is submaximal if and only if \(\mathcal{D}(\sigma)\in \operatorname{Deg}_{g,n}\) is submaximal (and realizable). The realizable submaximal elements of \(\operatorname{Deg}_{g,n}\) are of the following type:

• separating (mixed or unmixed), which are of the form \(\{(1;h,A),(1;g-h,A^c)\}\) and are realizable by Proposition 34;

• mixed (separating or not), which are of the form \(\{(e;h,A),(e;h,A)^c\}\) with \(\emptyset\subsetneq A \subsetneq [n]\) by Proposition 46. All such elements are realizable by Proposition 47;

• unmixed, non-separating and simple, which are of the form \(\{(e;h,\emptyset),(e;h,\emptyset)^c\}\) with \(\delta(e;h,\emptyset)=2h-2+e\geq g\) and \(e\geq 2\) by Corollary 10, Proposition 39[P:deg-grandi2] and Theorem 40[T:pos-Degn1b]. All such elements are realizable by Corollary 10 and Proposition 39[P:deg-grandi1].

• unmixed, non-separating and non-simple, which are of the form \(W_{\delta}\) for some \(1\leq \delta\leq g-1\) by Corollary 10 and Proposition 38. All such elements are realizable by Corollary 10 and by Theorem 40.

 ◻

Proof. If \(\sigma\in \Sigma_{g,n}^\chi\) is a submaximal element, then all the maximal elements of \(\Sigma_{g,n}^\chi\) that dominates \(\sigma\) are in bijection with the witnesses for \(\emptyset >\mathcal{D}(\sigma)\) by Corollary 6. Hence, we have to show that if \(D\) is a realizable submaximal element of \(\operatorname{Deg}_{g,n}\), then there are exactly two witnesses for \(\emptyset>D\). This is clear for simple submaximal elements and for the elements \(W_{\delta}\) it follows by combining Theorem 40[T:pos-Degn1a] and Proposition 45[P:reduceto1a]. ◻