January 01, 1970
The moduli spaces \(\mathcal{S}_g^-\) parametrise odd spin curves of genus \(g\). These are pairs \([C, \eta]\) where \(C\) is a smooth genus \(g\) curve of and \(\eta\) is a line bundle on \(C\) such that \(\eta^{\otimes 2} = \omega_C\) and \(h^0(C, \eta)\) is odd. The main result of this work is the tautology of the Chow ring of \(\mathcal{S}_5^-\). Our method of proof revolves around an analysis of the geometry of canonical genus 5 curves and totally tangent hyperplanes. In the course of establishing our main result, we also prove the rationality of the closely related differential stratum in \(\mathcal{M}_{5, 4}\) dominating \(\mathcal{S}_5^-\).
The moduli spaces \(\mathcal{S}_g\) parametrise objects of the form \([C, \eta]\) where \(C\) is a smooth curve of genus \(g\) and \(\eta \in \mathrm{Pic}(C)\) is a line bundle such that \(\eta^{\otimes 2} = \omega_C\). Such a a pair is called a spin curve. The forgetful map \(\pi : \mathcal{S}_g\rightarrow\mathcal{M}_g\) exhibits a direct connection between \(\mathcal{S}_g\) and \(\mathcal{M}_g\), and it is finite of degree \(2^{2g}\). Unlike \(\mathcal{M}_g\), the moduli spaces \(\mathcal{S}_g\) are not connected. In fact, \(\mathcal{S}_g\) has two irreducible components, \(\mathcal{S}_g^+\) and \(\mathcal{S}_g^-\), the moduli spaces of even (resp. odd) spin curves of genus \(g\) – see [1] and [2]. We say that a spin curve \([C, \eta]\) is even (resp. odd) if the number of sections \(h^0(C, \eta)\) is even (resp. odd).
In the landmark paper [3], Mumford began the study of the tautological ring \(\mathrm{R}^\bullet(\mathcal{M}_g)\) of \(\mathcal{M}_g\). Recall that \(\mathrm{R}^\bullet(\mathcal{M}_g)\) is the \(\mathbb{Q}\)–subalgebra of \(\mathrm{CH}^\bullet(\mathcal{M}_g)\) generated by the \(\kappa\)–classes \(\kappa_i := f_*(c_i(\omega_f)^{i +1})\), where \(f : \mathcal{C}_g \rightarrow\mathcal{M}_g\) is the universal curve and \(\omega_f\) is the relative dualising sheaf. Mumford then went on to ask whether the Chow ring \(\mathrm{CH}^\bullet(\mathcal{M}_g)\) could be tautological, at least in small genus. Regarding this question, significant progress has been made. With \(\mathbb{Q}\)–coefficients, one finds the following results:
Mumford determines the Chow ring \(\mathrm{CH}^\bullet(\overline{\mathcal{M}}_2)\) in [3].
Faber determines the Chow rings \(\mathrm{CH}^\bullet(\overline{\mathcal{M}}_3)\) and \(\mathrm{CH}^\bullet(\mathcal{M}_4)\) in [4] and [5].
Izadi determines the Chow ring \(\mathrm{CH}^\bullet(\mathcal{M}_5)\) in [6].
Vakil and Penev determine \(\mathrm{CH}^\bullet(\mathcal{M}_6)\) in [7].
Canning and H. Larson determine \(\mathrm{CH}^\bullet(\mathcal{M}_g)\) for \(g = 7, 8, 9\) in [8].
In all the works above, the key step is showing the tautology of the Chow ring which, using work by Faber in [9], allows one to compute the ideal of relations.
In contrast, much less is known about this problem for \(\mathcal{S}_g^\pm\). Define the tautological ring of \(\mathcal{S}_g^\pm\) (with rational coefficients) to be the subalgebra of \(\mathrm{CH}^\bullet(\mathcal{S}_g^\pm)\) generated by pullbacks of tautological classes on \(\mathcal{M}_g\) under the forgetful morphism \(\mathcal{S}_g^\pm \rightarrow\mathcal{M}_g\). The cohomological tautological ring of \(\mathcal{S}_g^\pm\) is the image under the cycle class map \(\mathrm{cl}: \mathrm{CH}^\bullet(\mathcal{S}_g^\pm) \rightarrow\mathrm{H}^\bullet(\mathcal{S}_g^\pm)\) of \(\mathrm{R}^\bullet(\mathcal{S}_g^\pm) \subset \mathrm{CH}^\bullet(\mathcal{S}_g^\pm)\).
Theorem 1. The moduli space \(\mathcal{S}_5^-\) of odd spin curves of genus \(5\) has tautological Chow and cohomology rings.
Notice that Theorem 1 in particular implies that \(\pi^* : \mathrm{CH}^\bullet(\mathcal{M}_5) \rightarrow\mathrm{CH}^\bullet(\mathcal{S}_5^-)\) is surjective so that, in fact, \(\pi^*\) is an isomorphism. Since the Chow ring of \(\mathscr{M}_5\) is known, cf. [6], we obtain the following explicit formula for the Chow ring of \(\mathcal{S}_5^-\).
Corollary 1. The Chow and cohomology ring of \(\mathcal{S}_5^-\) are given by \[\mathrm{CH}^\bullet(\mathcal{S}_5^-) \cong \mathrm{H}^\bullet(\mathcal{S}_5^-) \cong \mathbb{Q}[\kappa_1]/(\kappa_1^4).\]
Along the way towards the tautology of \(\mathrm{CH}^\bullet(\mathcal{S}_5^-)\), we also show a rationality result for the stratum of differentials \[\mathcal{M}_{5, 4}^{(2)}:= \{[C, p_1, \dots, p_4] \in \mathcal{M}_{5, 4}\colon \mathcal{O}_C(2p_1 + \dots + 2p_4) \cong \omega_C\},\] which is a \(\mathfrak{S}_4\)–cover of \(\mathcal{S}_5^-\).
Theorem 2. The differential stratum \(\mathcal{M}_{5, 4}^{(2)}\) is a rational variety.
Theorem 2 builds on [10] where Farkas and Verra, using Mukai models, provide (uni)rational parametrisations for moduli spaces of odd spin curves. Moreover, they complete the Kodaira classification of \(\mathcal{S}_g^-\) and we now know that \(\mathcal{S}_g^-\) is unirational for \(g \le 8\), uniruled for \(g \le 11\), and of general type for \(g \ge 12\). In our situation, the differential stratum \(\mathcal{M}_{5, 4}^{(2)}\) is proven to be rational after a studying the geometry of the restriction of nets of quadrics in \(\mathbb{P}^4\) to theta hyperplanes. This approach is manifestly dependent on the specific geometry of the general canonical curve of genus 5.
Throughout we work over \(\mathbb{C}\). All Chow and cohomology rings are with rational coefficients. We maintain the subspace convention for Grassmannians. We denote by \(X/G\) the quotient stack and by \(X/ \! \! /G\) the GIT quotient.
This subsection contains a reminder of a result by Vistoli which will be essential to our approach to the Chow ring \(\mathrm{CH}^\bullet(\mathcal{S}_5^-)\).
Theorem 3 (cf. [11]). Suppose \(G = \mathrm{GL}(r)\) or \(\mathrm{SL}(r)\) acts on a quasi-projective variety \(X\) and let \(\pi : X \rightarrow Y := X/G\) be the quotient. Denote by \(\mathcal{V}\rightarrow Y\) the vector bundle on \(Y\) corresponding to the principal \(G\)–bundle \(X \rightarrow Y\). The following hold:
The pullback \(\pi^* : \mathrm{CH}^\bullet(Y) \rightarrow\mathrm{CH}^\bullet(X)\) is surjective.
The kernel of \(\pi^*\) is generated as a group by classes of the form \(c_i(\mathcal{V}) \cap [Z]\) where \(i > 0\) and \([Z] \in \mathrm{CH}^\bullet(Y)\).
If \(C\) (hence \(Y\)) are smooth, then \(\mathrm{CH}^\bullet(Y)\) is generated as a ring by \(c_1(\mathcal{V}), \dots, c_r(\mathcal{V})\) and any chosen set of lifts of generators for \(\mathrm{CH}^\bullet(X)\).
This work was partially supported by the Deutsche Forsch-ungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, EXC-2046/2, project ID: 390685689).
I thank my advisor Gabi Farkas for his guidance and support throughout this project. I also thank Marian Aprodu, Riccardo Redigolo, Vlad Robu, Călin Spiridon and Andrei Stoenică for helpful conversations.
In this section we are going to discuss a birational model for the differential stratum \(\mathcal{M}_{5, 4}^{(2)}\) which parametrises smooth 4–pointed genus 5 curves \([C, p_1, \dots, p_4]\) such that \(\mathcal{O}_C(2p_1 + \dots + 2p_4) \cong \omega_C\). Using this model, Theorem 2 will then be proven. In our situation, this is a natural moduli space to study due to its relation to \(\mathcal{S}_5^-\), namely \(\mathcal{S}_5^-\approx \mathcal{M}_{5,4}^{(2)} / \mathfrak{S}_4\). This birational isomorphism simply maps \([C, p_1 + \dots + p_4] \in \mathcal{M}_{5, 4}^{(2)}/\mathfrak{S}_4\) to \([C, \mathcal{O}_C(p_1 + \dots + p_4)] \in \mathcal{S}_5^-\).
We start by motivating our construction. Let \([C, p_1, \dots, p_4] \in \mathcal{M}_{5, 4}^{(2)}\) be non–hyperellipic non–trigonal so that the canonical curve \(C \subset \mathbb{P}^4\) is a complete intersection of 3 quadrics \(Q_1, Q_2, Q_3\). The points \(p_1, \dots, p_4\) span a hyperplane \(H \subset \mathbb{P}^4\) which is totally tangent to \(C\) along the divisor \(p_1 + \dots + p_4\). Restricting the net of quadrics \(\Lambda := \langle Q_1, Q_2, Q_3\rangle\) to \(H \cong \mathbb{P}^3\), we obtain a net of quadrics in \(H\) which we denote by \(\Gamma\). The base locus of \(\Gamma\) is a zero dimensional subscheme of \(\mathbb{P}^3\) such that \(\mathrm{supp}\bigl(\mathrm{Bs}\,\Gamma\bigr) = p_1 + \dots + p_4\) and \(\mathrm{mult}_{p_i} \bigl(\mathrm{Bs}\,\Gamma\bigr) = 2\) for \(i = 1, \dots, 4\).
Let \(t_0, \dots, t_4\) be homogeneous coordinates on \(\mathbb{P}^4\) and embed \(\mathbb{P}^3\) in \(\mathbb{P}^4\) as the hyperplane \(H:= \{t_4 = 0\}\). The following variety is thus natural to study.
Definition 1. Fix \(p_1, \dots, p_4 \in \mathbb{P}^3\) in linearly general position. We let \(\mathbf{E}\subset \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) denote the locus of nets of quadrics \(\Gamma \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) such that \(\mathrm{supp}(\mathrm{Bs}\, \Gamma) = p_1 + \dots + p_4\), \(\mathrm{mult}_{p_i}(\mathrm{Bs}\,\Gamma) = 2\) for \(i = 1, \dots, 4\).
We define the rational map \(\rho\) to be the restriction map \[\rho : \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|) \dashrightarrow \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|), \quad \Lambda \mapsto \Lambda|_H.\] The exact structure of the fibres of \(\rho\) is readily identifiable. Let \(\Gamma \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) and notice that \(\rho^{-1}(\Gamma) \approx \mathbb{G}(2, \mathbb{P}V_\Gamma)\) where \(V_\Gamma\) is the vector space \(V_\Gamma := \Gamma \oplus t_4\cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1))\). Set \(\mathbb{C}^5 := t_4\cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1))\) and let \(\mathcal{S}\rightarrow\mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|)\) be the universal subbundle. We discover that there exists a birational isomorphism \[\begin{align} \label{eq:32key32grassmann32bd32struct32on32Grassmannian32of32nets32of32quadrics32in32P4} \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|) \approx G(3, \mathcal{S}\oplus (\mathbb{C}^5\times \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)) \end{align}\tag{1}\] over \(\mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\).
The exact locus of indeterminacy of \(\rho\) is easy to describe. The restriction of a net \(\Lambda \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) to \(H\) is undefined as a net of quadrics if and only if there exists a quadric \(Q \in \Lambda\) that contains \(H\), meaning \(\Lambda \cap t_4 \cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1)) \neq 0\). Therefore, the exact fibre of \(\rho\) over \(\Gamma\) is \[\begin{align} \label{eq:32fibre32of32rho32is32complement32of32Schubert32cycle} \rho^{-1}(\Gamma) = \{\Lambda \in G(3, V_\Gamma)\colon \Lambda \cap t_4\cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1)) = 0\}. \end{align}\tag{2}\] The following object is thus natural to investigate.
Definition 2. We define the subvariety \(\mathbf{F}\subset \mathbb{G}(2, \mathcal{O}_{\mathbb{P}^4}(2))\) to be the preimage of \(\mathbf{E}\) under the rational map \(\rho\), inside the domain of \(\rho\).
Recall the fixed points \(p_1, \dots, p_4 \in H \subset \mathbb{P}^4\) in linearly general position, and let \(\mathrm{P}G\subset \mathrm{PGL}(5)\) denote the stabiliser subgroup these points. In suitable coordinates, \[\mathrm{P}G \cong \left\{ \begin{bmatrix} * & 0 & 0 & 0 & *\\ 0 & * & 0 & 0 & * \\ 0 & 0 & * & 0 & * \\ 0 & 0 & 0 & * & * \\ 0 & 0 & 0 & 0 & * \end{bmatrix} \right\} \subset \mathrm{PGL}(5).\] Thus, we have arrived at the following birational model for \(\mathcal{M}_{5, 4}^{(2)}\).
Proposition 4. The map \(\mathbf{F}/ \! \! /\mathrm{P}G \dashrightarrow \mathcal{M}_{5, 4}^{(2)}\), \(\Lambda \mapsto [\mathrm{Bs}(\Lambda), p_1, \dots, p_4]\), is a birational isomorphism onto the locus of non–hyperelliptic, non–trigonal curves.
Building on Proposition 4, we now go on to tackle the problem of the rationality of the differential stratum \(\mathcal{M}_{5, 4}^{(2)}\).
Proof of Theorem 2. We first claim that \(\mathbf{F}\) is a rational variety. As it follows from (1 ), the restriction \(\rho|_\mathbf{F}: \mathbf{F}\rightarrow\mathbf{E}\) is a Grassmann bundle so the rationality of \(\mathbf{F}\) reduces to the rationality of \(\mathbf{E}\). This latter space is rational for the following reason.
Let \(\mathrm{P}G ' \subset \mathrm{PGL}(4) = \mathrm{Aut}(H)\) be the stabiliser group of the points \(p_1, \dots, p_4\) inside the automorphism group of the hyperplane \(H \cong \mathbb{P}^3\). This group identifies with a diagonal subgroup in \(\mathrm{PGL}(4)\), \(\mathrm{P}G' \cong (\mathbb{C}^*)^3\). Define \(\mathscr{X}\) to be the following incidence variety: \[\begin{align} \begin{tikzcd}[ampersand replacement=\&] \& {\mathscr{X}:= \{(P, \Gamma) \in \mathbb{G}\bigl(1, \bigl|\mathcal{I}_{\{p_i\}_{i=1}^4}(2)\bigr|\bigl) \times \mathbf{E}\colon P \subset \Gamma\}} \\ {\mathbb{G}\bigl(1, \bigl|\mathcal{I}_{\{p_i\}_{i=1}^4}(2)\bigl|\bigr)} \&\& \mathbf{E}, \arrow["{\pi_1}"', from=1-2, to=2-1] \arrow["{\pi_2}", from=1-2, to=2-3] \end{tikzcd} \end{align}\] where \(\pi_1 : \mathscr{X}\rightarrow\mathbb{G}(1, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) and \(\pi_2 : \mathscr{X}\rightarrow\mathbf{E}\) are the projections onto the first and second factors respectively.
We analyse \(\mathscr{X}\) more closely. Let \(P \in \mathbb{G}(1, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) be a general pencil of quadrics and set \(E := \mathrm{Bs}\, P\) for the base locus of \(P\). This is an elliptic curve of degree 4 in \(\mathbb{P}^3\). To give a net of quadrics \(\Gamma \in \mathbf{E}\) such that \(P \subset \Gamma\) is equivalent to giving a quadric totally tangent to \(E\). This shows that \(\pi_1\) is an isomorphism onto the locus of pencils of quadrics \(\Gamma\) for which, on the base locus \(E := \mathrm{Bs}(\Gamma)\), the line bundle \(\mathcal{O}_E(1)(-p_1 - \dots - p_4)\) has order 2 in \(\mathrm{Pic}^0(E)\). This analysis implies that there exists a birational isomorphism \[\begin{align} \mathscr{X}/ \! \! /\mathrm{P}G' \approx \mathcal{R}_{1;4} \end{align}\] where \(\mathcal{R}_{1;m}\) is the moduli space \[\mathcal{R}_{1;m}:= \{[E, p_1, \dots, p_m, \eta]\colon [E, p_1, \dots, p_m] \in \mathcal{M}_{1, m} \text{ and } \eta \in \mathrm{Pic}^0(E)[2]\text{ non--trivial}\}.\] The space \(\mathcal{R}_{1;4}\) is known to be (uni)rational so that, taking into account the induced projection \(\mathscr{X}/ \! \! /\mathrm{P}G' \rightarrow\mathbf{E}/ \! \! /\mathrm{P}G'\), we deduce that \(\mathbf{E}/ \! \! /\mathrm{P}G'\) is unirational as well.
We now point out that \(\pi_2 : \mathscr{X}\rightarrow\mathbf{E}\) is a \(\mathbb{P}^2\)–bundle, implying that \[\dim \mathbf{E}/ \! \! /\mathrm{P}G' = 2,\] that is, \(\mathbf{E}/ \mathrm{P}G'\) is a unirational surface. It is now a consequence of Castelnuovo’s Rationality Criterion that \(\mathbf{E}/ \! \! /\mathrm{P}G'\) is in fact rational. Since \(\mathbf{E}\rightarrow\mathbf{E}/ \! \! /\mathrm{P}G'\) is a principal \(\mathrm{P}G'\)–bundle in the étale topology and \(\mathrm{P}G' \cong (\mathbb{C}^*)^3\) is a simple group, this is a principal bundle in the Zariski topology. This proves the rationality of \(\mathbf{E}\), implying the rationality of \(\mathbf{F}\). By Myiata’s Theorem from [12] on the rationality of triangular representations, the rationality of \(\mathbf{F}/ \! \! /\mathrm{P}G\) follows. Coupled with Proposition 4, this finishes the proof of the rationality of \(\mathcal{M}_{5, 4}^{(2)}\). ◻
Motivated by the birational model discussed in Section 2 and for later use as well, the problem of this subsection is to exclude the possibility of obtaining by restriction nets of quadrics in \(\mathbb{P}^3\) with the following highly degenerate property: every subpencil of the net has singular base locus. Let \([C, \eta] \in \mathcal{S}_5^-\) be a non–hyperelliptic non–trigonal odd spin curve of genus 5 with \(H \subset \mathbb{P}^3\) the hyperplane giving rise \(\eta\) on \(C\). Write \(\eta = \mathcal{O}_C(p_1 + \dots + p_4)\) for not necessarily distinct points \(p_1, \dots, p_4\in C\). Let \(\Lambda \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|)\) be the net of quadrics whose complete intersections is \(C\) and \(\Lambda_H:= \Lambda|_H\) be the restriction to \(H\).
Segre’s classification of pencils of quadrics tells us in particular that if \(P\) has Segre symbol \([1, 1, 1, 1]\) (meaning that the discriminant polynomial \(\mathrm{disc}\, P\) has no multiple roots), the base locus \(\mathrm{Bs}\, P\) is smooth. Therefore, we must discard two possibilities according to the discriminant \(\Delta := \mathrm{disc}\, \Lambda_H\):
\(\Delta = |\Lambda_H|\), i.e. every quadric in \(\Lambda_H\) is singular. As it turns out, this is equivalent to the net \(\Lambda_H\) failing to be semistable ([13]).
\(\Delta\) has multiple components.
In discarding these two cases, the following lemma will shortly prove its usefulness.
Lemma 1. No two quadrics in \(\Lambda_H\) share a common singularity amongst the points \(p_i\).
Proof. Suppose there exist quadrics \(q_i, q_i' \in \Lambda_H\) linearly independent both singular at \(p_i\), and let \(P := \langle q_i, q_i' \rangle \subset \Lambda_H\) be the pencil they generate. Let \(\widetilde{P} \subset \Lambda\) be any lift of the pencil \(P\), and consider the vector space \[V_{p_i}:= \{\ell(p_i) = 0\} \subset \Lambda_H \oplus t_4 \cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1)).\] This is the hyperplane of all quadrics in \(Q \in \Lambda_H \oplus t_4 \cdot \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1))\) which, if written \(Q = q + t_4 \cdot \ell\) for \(q \in \Lambda_H\) and \(\ell \in \mathrm{H}^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(1))\), \(\ell(p_i) = 0\) holds. Let \(Q_0 \in \widetilde{P} \cap V_{p_i}\) be non–zero (which exists for dimension reasons). One checks easily that \(Q_0\) is singular at \(p_i\) so that we have found a quadric \(Q_0 \in \Lambda\) which is singular at \(p_i \in \mathrm{Bs}\, \Lambda\). This implies that \(C = \mathrm{Bs}\, \Lambda\) is singular at \(p_i\), a contradiction. ◻
We now turn our attention back to cases \(i)\) and \(ii)\). Possibility \(i)\) is easy to discard if one recalls Wall’s classification of non–semistable nets of quadrics in \(\mathbb{P}^3\) as shown in (3 ). \[\begin{align} \label{eq:32non--semistable32nets32of32quadrics32in32P3} \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & * & * & * \\ 0 & * & * & * \\ 0 & * & * & * \end{pmatrix} \quad \text{or} \quad \begin{pmatrix} 0 & 0 & 0 & * \\ 0 & 0 & 0 & * \\ 0 & 0 & * & * \\ * & * & * & * \end{pmatrix}. \end{align}\tag{3}\] In the second matrix, the base locus contains a line and this cannot happen since \(\dim \mathrm{Bs}(\Lambda_H) = 0\). As for the first matrix, \([1:0:0:0] \in \mathrm{Bs}\, \Lambda_H\) is a singularity shared by all of the quadrics in the net, contradicting Lemma 3.
We can now assume that \(\Delta \neq |\Lambda_H|\), and we want to analyse what happens when \(\Delta\) is non–reduced. The following lemma follows from [13]
Lemma 2. Let \(\Gamma \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^3}(2)|)\) be a net of quadrics. If \(\Gamma\) is stable, the discriminant \(\Delta\) has a multiple component if and only if \(\dim (\mathrm{Bs}\,\Gamma) > 0\).
We are thus left to deal with unstable nets of quadrics in \(\mathbb{P}^3\) which are semistable. Namely, we have the following possibilities (cf. [13]) for all matrices in the given net: \[\begin{pmatrix} 0 & 0 & 0 & * \\ 0 & * & * & * \\ 0 & * & * & * \\ * & * & * & * \end{pmatrix} \quad \text{or} \quad \begin{pmatrix} 0 & 0 & * & * \\ 0 & 0 & * & * \\ * & * & * & * \\ * & * & * & * \end{pmatrix}.\] Notice again that the second type of matrix is not possible by dimension reasons as the base locus of such a net contains a line.
We now take a closer look at the first type. From Wall’s classification [13] of semistable nonstable quadrics, only cases (i) and (ii) are worth considering as otherwise the base locus contains a component of dimension \(>0\). Take for example (i). This is the scenario where all quadrics are tangent to a fixed plane \(\pi \subset \mathbb{P}^3\) at a point \(x_1 \in \pi\). Let now \(P\) be any pencil in such a net. Since \(x_1 \in \mathrm{Bs}\, P\) is a singularity of \(\mathrm{Bs}\, P\) (the quadrics share a common tangent plane there), there exists a quadric \(q \in P\) singular at \(x_1\). Picking a different pencil \(P'\) in the net which avoids \(q\), we find another quadric \(q' \in P'\) which is also singular at \(x_1\), contradicting Lemma 3. Case (ii) is dealt with by a similar token.
The discussion in this subsection has the following useful consequence.
Lemma 3. Let \([C, \eta] \in \mathcal{S}_5^-\), \(\eta = \mathcal{O}_C(D)\), be non–hyperelliptic, non–trigonal and view \(C\) canonically embedded in \(\mathbb{P}^4\). Let \(H \in (\mathbb{P}^4)^\vee\) be such that \(C \cdot H = 2D\). The discriminant \(\Delta := \mathrm{disc}\, \Lambda|_H\) is not the entire net \(|\Lambda|\), nor does it have multiple components. In particular, the general pencil \(P \subset |\Lambda|\) has base locus a smooth elliptic curve.
Definition 3. Let \(\widetilde{\mathcal{S}}_5^-\subset \mathcal{S}_5^-\) denote the locus of non–hyperelliptic, non–trigonal curves with reduced theta characteristic.
The main object of interest in this section is the open subset \(\widetilde{\mathcal{S}}_5^-\) in \(\mathcal{S}_5^-\). Using a variant of the model described in Section 2, we prove the following proposition.
Proposition 5. The Chow and cohomology rings of \(\widetilde{\mathcal{S}}_5^-\) is generated by tautological classes.
Before going into the proof, we define some of the varieties that will be necessary along the way. Consider the following incidence variety: \[\begin{align} \mathscr{F}:= \left\{(\Lambda, H) \in \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|) \times (\mathbb{P}^4)^\vee \colon \begin{matrix} \mathrm{supp}(\mathrm{Bs}\, \Lambda|_H) = p_1 + \dots + p_4, \text{ p_i \in H are} \\ \text{distinct, } \mathrm{mult}_{p_i}(\mathrm{Bs}\, \Lambda|_H) = 2 \text{ for }i = 1, \dots, 4 \end{matrix}\right\}. \end{align}\] This is just the version of the variety \(\mathbf{F}\) in Definition 2 where we allow both the points \(p_i\) in the support of \(\mathrm{Bs}(\Lambda|_H)\) as well as the hyperplane \(H\) to vary. Consider the two projections \(\mathrm{pr}_1 : \mathscr{F}\rightarrow\mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|)\) and \(\mathrm{pr}_2 : \mathscr{F}\rightarrow(\mathbb{P}^4)^\vee\) onto the first and second factors. Notice that \(\mathrm{pr}_1\) is finite as the fibre \(\mathrm{pr}_1^{-1}(\Lambda)\) parametrises totally tangent hyperplanes to \(C := \mathrm{Bs}(\Lambda)\). It follows that \[\dim \mathscr{F}= \dim \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|) = \dim G(3, 15) = 36.\] The group \(\mathrm{PGL}(5)\) acts on \(\mathscr{F}\subset \mathbb{G}(2, |\mathcal{O}_{\mathbb{P}^4}(2)|) \times (\mathbb{P}^4)^\vee\) and we find the following birational model for \(\mathcal{S}_5^-\). Let \(\mu : \mathscr{F}/ \mathrm{PGL}(5) \dashrightarrow \mathcal{S}_5^-\) be the map \[[\Lambda, H] \mapsto [C, \eta] \quad \mathrm{for }C = \mathrm{Bs}\, \Lambda\, , \eta = \mathcal{O}_C(p_1 + \dots + p_4)\] where \(C \cdot H = 2p_1 + \dots + 2p_4\). The following result now holds:
Proposition 6. The map \(\mu : \mathscr{F}/ \mathrm{PGL}(5) \dashrightarrow \mathcal{S}_5^-\) is a a birational isomorphism onto the locus \(\widetilde{\mathcal{S}}_5^-\).
Proof of Proposition 5. We start by observing that \(\mathscr{F}/ \mathrm{SL}(5) \rightarrow\mathscr{F}/\mathrm{PGL}(5)\) is a \(\mu_5\)–banded gerbe so that, since we are working with rational coefficients, we obtain an isomorphism of Chow rings \(\mathrm{CH}^\bullet(\mathscr{F}/\mathrm{PGL}(5)) \cong \mathrm{CH}^\bullet(\mathscr{F}/\mathrm{SL}(5))\). Let \(\mathcal{V}\rightarrow\mathscr{F}/\mathrm{SL}(5)\) be the vector bundle induced by \(\mathscr{F}\rightarrow\mathscr{F}/\mathrm{SL}(5)\) so that \(c_1(\mathcal{V}) = 0\). By Theorem 3, the Chow ring \(\mathrm{CH}^\bullet(\mathscr{F}/\mathrm{SL}(5))\) is generated as a \(\mathbb{Q}\)–algebra by the Chern classes \(\{c_i(\mathcal{V})\}_{i = 2}^5\) and any lift of generators for \(\mathrm{CH}^\bullet(\mathscr{F})\). The same type of argument as in [7] gives that \(c_i(\mathcal{V})\) are all polynomials in the \(\lambda\)–classes. We next want to focus on \(\mathrm{CH}^\bullet(\mathscr{F})\).
The key auxiliary variety needed for our current goal is the following analogue of the variety \(\mathbf{E}\) from Section 2. Let \(\mathcal{W}\rightarrow(\mathbb{P}^4)^\vee\) be the universal bundle so that \(\mathbb{P}\mathcal{W}\rightarrow(\mathbb{P}^4)^\vee\) is the universal hyperplane, and let \(\pi: G(3,\mathrm{Sym}^2(\mathcal{W}^\vee)) \rightarrow(\mathbb{P}^4)^\vee\) be the Grassmann bundle of 3–planes in \(\mathrm{Sym}^2(\mathcal{W}^\vee)\). Define \[\begin{align} \label{eq:32definition32of32Bscr} \mathscr{B}:= \left\{\Gamma \in G(3, \mathrm{Sym}^2(\mathcal{W}^\vee)) \colon \begin{matrix} \text{if H := \pi(\Gamma), \mathrm{supp}\, \mathrm{Bs}(\Gamma) = p_1 + \dots + p_4 ,} \\ \text{then }\mathrm{mult}_{p_i}(\mathrm{Bs}\, \Gamma) = 2 \text{ for }i = 1, \dots, 4 \end{matrix}\right\}. \end{align}\tag{4}\] As a subvariety of \(G(3, \mathrm{Sym}^2(\mathcal{W}^\vee))\), \(\mathscr{B}\) admits a projection \(\mathscr{B}\rightarrow(\mathbb{P}^4)^\vee\). Define \(\varrho\) to be the restriction map \(\varrho : \mathscr{Y}\rightarrow\mathscr{B}\), given by \[\varrho : (\Lambda, H) \mapsto \Lambda|_H \in \mathbb{G}(2, |\mathcal{O}_H(2)|).\] As in Section 2 where the hyperplane \(H\) and the points \(p_1, \dots, p_4\) were fixed, we see that \(\varrho : \mathscr{F}\rightarrow\mathscr{B}\) is a Zariski-open subset of a Grassmann bundle over \(\mathscr{B}\). Thus, just as in (1 ) and (2 ), we find that \(\mathscr{F}\) is the complement of a Schubert cycle of complementary type, which is a linear section in the Plücker embedding of the Grassmann bundle containing \(\mathscr{Y}\). The Andreotti–Frankel Vanishing Theorem for the cohomology of affine varieties (which holds for Chow rings of Grassmann bundles) implies that we have an isomorphism \(\mathrm{CH}^\bullet(\mathscr{B}) \cong \mathrm{CH}^\bullet(\mathscr{F})\).
We now shift our attention to \(\mathscr{B}\subset G(3, \mathrm{Sym}^2(\mathcal{W}^\vee)) \rightarrow(\mathbb{P}^4)^\vee\). Consider the incidence variety \(\mathscr{A}\) given by \[\begin{align} \label{eq:32key32incidence32variety32Ascr} \mathscr{A}:= \{(P, \Gamma) \in G(2, \mathrm{Sym}^2(\mathcal{W}^\vee)) \times_{(\mathbb{P}^4)^\vee}\mathscr{B}\colon P \subset \Gamma\} \end{align}\tag{5}\] and the two projections \[\begin{tikzcd} & \mathscr{A}\\ {G(2, \mathrm{Sym}^2(\mathcal{W}^\vee))} && {\mathscr{B}} \arrow["{q_1}"', from=1-2, to=2-1] \arrow["{q_2}", from=1-2, to=2-3] \end{tikzcd}.\] The projection \(q_2 : \mathscr{A}\rightarrow\mathscr{B}\) is a \(\mathbb{P}^2\)–bundle with fibre \(q_2^{-1}(\Gamma) = G(2, \Gamma)\) which by the Projective Bundle Formula gives that \(\mathrm{CH}^\bullet(\mathscr{B}) = \mathrm{CH}^\bullet(\mathscr{A}) / \langle c_1(\mathcal{O}_\mathscr{A}(1))\rangle\). The Chow ring of \(\mathscr{A}\) is understood by relating \(\mathscr{A}\) to \(\mathcal{R}_{1;[4]}\). Namely, just as in Subsection 2, we find the birational isomorphism \(\mathscr{A}/\mathrm{PGL}(5) \xrightarrow[]{\approx} \mathcal{R}_{1;[4]}\). The Chow ring of \(\mathcal{R}_{1;4}\) is known to be generated by the \(\psi\)–classes by [14], giving that the Chow ring of \(\mathcal{R}_{1;[4]}\) is generated by symmetric polynomials in them. In particular, the Chow ring of \(\mathscr{A}/\mathrm{PGL}(5) \approx \mathcal{R}_{1;4}\) is generated by symmetric polynomials in (pullbacks of) the \(\psi\)–classes. Vistoli’s Theorem 3 now gives that \(\mathrm{CH}^\bullet(\mathscr{A})\) is generated by the \(\psi\)–classes as well.
Tracing back through the arguments so far we find that \(\mathrm{CH}^\bullet(\widetilde{\mathcal{S}}_5^-)\) is generated by symmetric polynomials in the \(\psi\)–classes coming from the identification \(\mathcal{M}_{5, 4}^{(2)}/\mathfrak{S}_4 \cong \mathcal{S}_5^-\). Witney’s Formula together with the relation \(\eta^{\otimes 2} = \omega\) on the universal curve over \(\mathcal{S}_5^-\) finally allow us to express the \(\psi\)–classes in terms of the \(\lambda\)–classes. This finishes the proof of the Proposition. ◻
Let \(\mu = (2\mu_1, \dots, 2\mu_n)\) be a partition of \(8\) where \(\mu_i > 0\) for all \(i\). We denote by \(\widetilde{\mathcal{S}}_5^-(\mu) \subset\mathcal{S}_5^-\) the image of the non–hyperelliptic, non–trigonal locus in the differential stratum \[\mathcal{M}_{5, n}^\mu := \{[C, p_1, \dots, p_n] \in \mathcal{M}_{5, n}\colon \mathcal{O}_C(2\mu_1p_1 + \dots + 2\mu_np_n) \simeq \omega_C\}.\] In this subsection, we explain how almost identical arguments to the proof of Proposition 5 yield that \(\mathrm{CH}^\bullet\bigl(\widetilde{\mathcal{S}}_5^-(\mu)\bigr)\) yield the following theorem.
Proposition 7. The Chow and cohomology rings of the stratum \(\widetilde{\mathcal{S}}_5^-(\mu)\) are generated by restrictions of tautological classes for any partition \(\mu = (2\mu_1, \dots, 2\mu_n)\) of 8 such that \(\mu_i > 0\) for all \(i\). Moreover, all the classes \(\bigl[\widetilde{\mathcal{S}}_5^-(\mu)\bigr]\) are tautological inside the Chow ring of the non–hyprelliptic, non–trigonal locus \(\widetilde{\mathcal{S}}^-_5 \setminus (\mathcal{S}\mathcal{H}_5^- \cup \mathcal{S}\mathcal{T}_5^-)\).
Proof. We start by addressing the first claim. Consider the following generalisation of (4 ) for all even partitions \(\mu = (2\mu_1, \dots, 2\mu_n)\) of 8: \[\begin{align} \mathscr{B}(\mu) := \left\{\Gamma \in G(3, \mathrm{Sym}^2(\mathcal{W}^\vee)) \colon \begin{matrix} \text{if H := \pi(\Gamma), }\mathrm{supp}(\mathrm{Bs}\, \Gamma) = p_1+\dots+ p_n, \\ \text{then }\mathrm{mult}_{p_i}(\mathrm{Bs}\, \Gamma) = \mu_i\text{ for } i = 1, \dots, n \end{matrix}\right\}. \end{align}\] Define the incidence variety \(\mathscr{A}(\mu)\) by \[\begin{align} \mathscr{A}(\mu) := \{(P, \Gamma) \in G(2, \mathrm{Sym}^2(\mathcal{W}^\vee)) \times_{(\mathbb{P}^4)^\vee}\mathscr{B}(\mu) \colon P \subset \Gamma\} \end{align}\] By Lemma 3, the open subset \[\mathcal{U}(\mu) := \{(P, \Gamma) \in \mathscr{A}(\mu)\colon \mathrm{Bs}\, P\text{ is smooth}\} \subset \mathscr{A}(\mu)\] surjects onto \(\mathscr{B}(\mu)\). We are thus reduced to showing that the Chow ring of \(\mathcal{U}\) is generated by tautological classes. This is done almost identically to Subsection 4, with the following differences:
\(\mathcal{U}(2, 2, 4)/ \! \! /\mathrm{PGL}(5) \cong \mathcal{R}_{1;1+[2]} := \mathcal{R}_{1;4}/\mathfrak{S}_2\) where \(\mathfrak{S}_2\) permutes the last two marked points.
\(\mathcal{U}(4, 4)/ \! \! /\mathrm{PGL}(5) \cong \mathcal{R}_{1;2}\).
\(\mathcal{U}(8) / \! \! /\mathrm{PGL}(5) \cong \mathcal{R}_{1;1}\).
As in our study of the locus of spin curves with reduced theta characteristics, we see that all the loci \(\widetilde{\mathcal{S}}_5^-(\mu)\) have tautological Chow ring by results in [14].
We now claim that the classes \([\widetilde{\mathcal{S}}_5^-(\mu)] \in \mathrm{R}^\bullet(\widetilde{\mathcal{S}}\) Consider the partial compactification \(\mathcal{M}_{g, g-1}^{(2)}\) inside the \((g-1)\)–fold universal curve \(\mathcal{C}_g^{g - 1} = \mathcal{C}_g \times_{\mathcal{M}_g} \dots \times_{\mathcal{M}_g} \mathcal{C}_g\): \[\widehat\mathcal{M}_{g, g-1}^{(2)}:= \left\{[C, p_1, \dots, p_{g - 1}] \in \mathcal{C}_g^{g - 1}\colon \begin{matrix} \mathcal{O}_C(2p_1 + \dots + 2p_{g - 1}) = \omega_C \end{matrix}\right\},\] which dominates \(\mathcal{S}_g^-\). The diagonals in \(\mathcal{C}_5^4\) cut on \(\widehat\mathcal{M}_{5, 4}^{(2)}\) loci which dominate \(\mathcal{S}_5^-(\mu)\) so that the corresponding locus in \(\widehat \mathcal{M}_{5, 4}^{(2)}\) pushes forward to a non–zero multiple of the stratum \(\mathcal{S}_5^-(\mu)\) for all even positive partitions \(\mu\) of 8. This implies the tautology of the classes \(\bigl[\mathcal{S}_5^-(\mu)\bigr]\). Restricting to the open locus \(\widetilde{\mathcal{S}}_5^-\), we obtain the proposition. ◻
Using push and pull, the results of this section are summarised in the following corollary.
Corollary 2. The Chow ring \(\mathrm{CH}^\bullet(\mathcal{S}_5^-\setminus(\mathcal{S}\mathcal{T}_5^- \cup \mathcal{S}\mathcal{H}_5^-))\) of the locus of non–hyprelliptic, non–trigonal spin curves of genus 5 is generated by tautological classes.
By Corollary 2, we are left to analyse the hyperelliptic and trigonal loci.
Definition 4.
We denote by \(\mathcal{H}_g \subset \mathcal{M}_g\) the hyperelliptic locus in \(\mathcal{M}_g\) and by \(\mathcal{S}\mathcal{H}_g \subset \mathcal{S}_g\) the trigonal locus to \(\mathcal{S}_g\).
We denote by \(\mathcal{T}_g \subset \mathcal{M}_g\) the trigonal locus in \(\mathcal{M}_g\) and by \(\mathcal{S}\mathcal{T}_g^\pm \subset \mathcal{S}_g^\pm\) the pullback of the trigonal locus to \(\mathcal{S}_g^\pm\).
We begin by recalling the following well–known description of theta characteristics on a hyperelliptic curve (cf. [15]).
Proposition 8. Let \(\alpha : C \rightarrow\mathbb{P}^1\) be a smooth hyperelliptic curve of genus \(g\) with hyperelliptic linear system \(|D| = |\alpha^*\mathcal{O}_{\mathbb{P}^1}(1)|\). Let \(x_1 + \dots + x_{2g + 2}\) be the ramification divisor of \(\alpha\).
Any theta characteristic on \(C\) is of the form \(\eta = \mathcal{O}_C(E)\) where \[E = kD + x_{i_1} + \dots + x_{i_{g - 1 - 2k}}\] for some \(-1 \leq k \leq \left\lfloor\frac{g - 1}{2}\right\rfloor\).
The expression of \(\eta\) in i) is unique if \(k \geq 0\). If \(k = -1\), then \[-D + x_{i_1} + \dots + x_{i_{g + 1}} \sim -D + x_{j_1} + \dots + x_{j_{g + 1}}\] if and only if \(\{{i_1}, \dots, {i_{g+1}}\} \cup \{{j_1}, \dots, {j_{g + 1}}\} = \{1, \dots, 2g + 2\}\).
If a theta characteristic \(\eta\) has the form as in i), then \(h^0(C, \eta) = k + 1\).
Proposition 8 has the following immediate consequence. Consider the loci \[\begin{align} \mathcal{S}\mathcal{H}_g(k) := \{[C, \eta] \in \mathcal{S}\mathcal{H}_g\colon h^0(C, \eta) = k + 1\} \end{align}\] for \(-1 \le k \le \lfloor\frac{g -1}{2}\rfloor\). The following corollary is now clear.
Corollary 3. The irreducible components of the hyperelliptic locus \(\mathcal{S}\mathcal{H}_g\) are the subvarieties \(\mathcal{H}{\mathcal{S}}_g(k)\) for \(-1 \leq k \leq \lfloor\frac{g - 1}{2}\rfloor\).
We construct, for each \(-1 \leq k \leq \lfloor\frac{g -1}{2}\rfloor\), a finite étale map \(a_k :\mathcal{M}_{0, 2g + 2} \rightarrow\mathcal{S}\mathcal{H}_g(k)\) which sends \([\mathbb{P}^1, y_1, \dots, y_{2g + 2}] \mapsto [C, \eta]\), where \(\alpha : C \rightarrow\mathbb{P}^1\) is the unique double cover of \(\mathbb{P}^1\) branched at \(y_1 + \dots + y_{2g + 2}\) and, setting \(\{x_i\} = \alpha^{-1}(y_i)\), \(\eta = \mathcal{O}_C(kD + x_1 + \dots + x_{g-1 - 2k})\). These maps induce isomorphisms \[\begin{align} \label{eq:32concrete32description32SH95g40k41} \mathcal{S}\mathcal{H}_g(k) \cong \mathcal{M}_{0, 2g + 2} / \mathfrak{S}_{g - 1 - 2k} \rtimes \mathfrak{S}_{g + 3 +2k}. \end{align}\tag{6}\] From (6 ) we deduce that the \(\mathcal{S}\mathcal{H}_g(k)\) is Chow–free.
With all this background in mind, we can now tackle the main theorem of this subsection. The following holds:
Theorem 9. The loci \(\mathcal{S}\mathcal{H}_g(k)\) are all Chow–free. Moreover, the classes \([\mathcal{S}\mathcal{H}_{5}(k)] \in \mathrm{CH}^\bullet(\mathcal{S}_{5}^-)\) are tautological for \(k = 0, 2\).
Proof. Only the tautology of the classes \([\mathcal{S}\mathcal{H}_5(k)]\) for \(k = 0, 2\) remains to be addressed. Since the hyperelliptic locus in \(\mathcal{M}_g\) is tautological, it suffices to show \([\mathcal{S}\mathcal{H}_5(2)] \in \mathrm{R}^\bullet(\mathcal{S}_5^-)\). Notice that \(\mathcal{S}\mathcal{H}_5(2)\) can be written as the locus \[\mathcal{S}\mathcal{H}_5(2) = \{[C, \eta]\in \mathcal{S}_5^-\colon h^0(C, \eta) \ge 3\}.\] This identification follows by Clifford’s Theorem and Proposition 8. Let \(d \gg 0\) and \(p \in C\). Consider the cohomology sequence \[0 \rightarrow\mathrm{H}^0(C, \eta) \rightarrow\mathrm{H}^0(C, \eta(dp)) \xrightarrow[]{\mathrm{ev}(C, p, \eta)} \mathrm{H}^0(C, \eta(dp)/\eta) \rightarrow\mathrm{H}^1(C, \eta) \rightarrow 0.\] This picture of course globalises, and we are interested in loci of the form \[\{[C, p, \eta]\colon\dim \mathrm{Ker}\left(\mathrm{ev}(C, p, \eta)\right) \ge k + 1\}.\] We now express the preimage of \(\mathcal{S}\mathcal{H}_5(2)\) in \(\mathcal{S}_{5;1}^- := \mathcal{S}_5^-\times_{\mathcal{M}_5} \mathcal{M}_{5, 1}\) as a symmetric degeneracy locus. Let \((f : \mathcal{C}_{g;1}^- \rightarrow\mathcal{S}_{g;1}^-, \sigma : \mathcal{S}_{g;1}^- \rightarrow\mathcal{C}_{g,1}^-)\) be the universal curve over \(\mathcal{S}_{g;1}^- = \mathcal{S}_g^- \times_{\mathcal{M}_g} \mathcal{M}_{g;1}\). Let \(\eta\) be the universal theta characteristic on \(\mathcal{C}_{g;1}^-\). Consider the long exact higher direct image sequence \[0 \rightarrow f_*\eta \rightarrow f_*\eta(d\sigma) \xrightarrow[]{\mathrm{ev}} f_*\left[\eta(d\sigma) / \eta\right] \rightarrow\mathrm{R}^1f_*\eta \rightarrow 0.\] Set \(\mathcal{A}:= f_*\eta(d\sigma)\) and \(\mathcal{B}:= f_*(\eta \otimes \mathcal{O}_{d\sigma}(d\sigma))\). The subvariety \[\mathcal{S}_g^-(k) := \{[C, p, \eta]\colon h^0(C, \eta) \ge k + 1\}\] for \(k\) even is the degeneracy locus of \(\mathrm{ev}\). We claim that \(\mathcal{B}\cong \mathcal{A}^\vee\). Indeed, we have that, by Serre duality, \[\mathcal{B}^\vee = (f_*(\eta(d\sigma)/\eta))^\vee \cong \mathrm{R}^1f_*(\omega_f \otimes (\eta\otimes \mathcal{O}_{d\sigma}(d\sigma))^\vee) = \mathrm{R}^1f_*(\eta \otimes \mathcal{O}_{d\sigma}(-d\sigma)).\] We examine closer the rank \(d\) vector bundle \(\mathrm{R}^1f_*(\eta \otimes \mathcal{O}_{d\sigma}(-d\sigma))\). Consider the short exact sequence \[0 \rightarrow\eta(-2d\sigma) \rightarrow\eta(-d\sigma) \rightarrow\eta \otimes \mathcal{O}_{d \sigma}(-d\sigma) \rightarrow 0,\] giving rise to the map of vector bundles \[\mathrm{R}^1f_*\eta(-d\sigma) \rightarrow\mathrm{R}^1f_*(\eta \otimes \mathcal{O}_{d\sigma}(-d\sigma)) \rightarrow 0.\] Notice that the fibres of \(\mathrm{R}^1f_*\eta(-d\sigma)\) are \[(\mathrm{R}^1f_*\eta(-d\sigma))_{[C, p]} = \mathrm{H}^1(C, \eta_C(-dp)),\] and we compute \(h^1(C, \eta_C(-dp)) = h^0(C, \eta_C(dp)) = d\). Therefore, the homomorphism \(\mathrm{R}^1f_*\eta(-d\sigma) \rightarrow\mathrm{R}^1f_*(\eta\otimes \mathcal{O}_{d\sigma}(-d\sigma))\) is a surjection between vector bundles of the same rank, hence an isomorphism. Thus, we conclude that \(\mathcal{A}\cong \mathcal{B}^\vee\).
We can now apply [16] if \(\mathcal{S}_g^-(k)\) has expected codimension \(\binom{k + 1}{2}\). In our situation, we have that \(\mathcal{S}_5^-(2)= \mathcal{S}\mathcal{H}_5(2)\) has expected codimension 3. It follows that the class \([\mathcal{S}\mathcal{H}_5(2)]\) is tautological. ◻
The following is the main result of thie subsection.
Proposition 10. The locus \(\mathcal{T}\mathcal{S}_5^-\) has Chow ring generated by restrictions of tautological classes from \(\mathcal{S}_5^-\). Moreover, the class \([\mathcal{T}\mathcal{S}_g^-]\) is tautological for any \(g\).
Before going into the proof, we recall the well known interpretation of genus 5 trigonal curves as plane quintics with one singularity which is either a node or a cusp. More precisely, let \(C\) be a non–hyperelliptic trigonal curve of genus 5 and \(D\) a trigonal divisor. The linear system \(|K_C - D|\) gives rise to a birational map \(\varphi : C \rightarrow\mathbb{P}^2\) such that \(\Gamma := \mathrm{Im}\,\varphi\) is a plane quintic with one singularity of \(\delta\)–invariant 1 by the genus formula. If \(\{p_0\} =\mathrm{Sing}(\Gamma)\) and \(\varphi^{-1}(p_0) = \{p_0^+,p_0^-\}\), the points \(p_0^+\), \(p_0^-\) are distinct if \(p_0\) is a node, and \(p_0^+ = p_0^-\) if \(p_0\) is a cusp. Notice that the divisor \(p_0^+ + p_0^-\) is the unique divisor in \(|K_C -2D|\).
Proof of Theorem 10. To prove that \(\mathrm{CH}^\bullet(\mathcal{T}\mathcal{S}_5^-)\) generated by restrictions of tautological classes from \(\mathcal{S}_5^-\), we split the trigonal locus into the following locally closed strata:
\(\mathcal{T}^{(1)}\): The locus of pairs \([C, \eta]\) such that \(K_C - 2D \sim 2p_0^+\) for some \(p_0^+ \in C\) and \(h^0(C, \eta(-p_0^+)) \neq 0\).
\(\mathcal{T}^{(2)}\): The complement of \(\mathcal{T}^{(1)}\) in \(\mathcal{T}\mathcal{S}_5^-\).
We study first \(\mathcal{T}^{(2)}\) which is a dense open subset of \(\mathcal{T}\mathcal{S}_5^-\). Let \([C, \eta] \in \mathcal{T}^{(2)}\) and maintain the notation in the first paragraph of this subsection with the convention that \(p_0^+ = p_0^-\) if \(\Gamma\) has a cusp at \(p_0\). For parity reasons and also taking into account that \(p_0^+ \notin \mathrm{supp}\, \eta\) it follows that \(\eta\) is given by a smooth conic through \(p_0\) which is tangent to \(\Gamma\) at four other points.
Consider thus the following incidence variety: \[\begin{align} \mathscr{Q}^{(2)} := \{(\Gamma, p_0, \dots, p_4)\colon \Gamma \in |\mathcal{O}_{\mathbb{P}^2}(5)| \text{ is singular at p_0, }\Gamma\cdot C_\mathbf{p}= 2p_0 + \dots + 2p_4\}. \end{align}\] In this context, if \(\mathbf{p}= (p_0, \dots, p_4)\) then \(C_\mathbf{p}\) stands for the unique conic through the five points \(p_0, \dots, p_4\). Let \(\pi : \mathscr{Q}^{(2)} \rightarrow(\mathbb{P}^2)^5\) be the projection remembering the points \((p_0, \dots, p_4)\) and consider the open locus \(\mathcal{U}\subset (\mathbb{P}^2)^5\) of tuples \(\mathbf{p}\) such that \(C_\mathbf{p}\) is smooth. We obtain a birational isomorphism \[\begin{align} \mathcal{T}^{(2)} \approx (\mathscr{Q}^{(2)}/ \! \! /\mathrm{PGL}(3))/\mathfrak{S}_4. \end{align}\] This exhibits \(\mathcal{T}^{(2)}\) as an open subset of \((\mathscr{Q}^{(2)}/ \! \! /\mathrm{PGL}(3))/\mathfrak{S}_4\). Methods very similar to the ones described for the loci \(\widetilde{\mathcal{S}}_5^-\) allow us to see that \(\mathrm{CH}^\bullet(\mathcal{T}^{(2)})\) has Chow ring generated by restrictions of tautological classes from \(\mathcal{S}_5^-\).
We next continue to study \(\mathcal{T}^{(1)}\). In this case, \(\eta = \mathcal{O}_C(p_0^+ + p_1 + p_2 + p_3)\) where \(p_1 + p_2 + p_3\) is a divisor on \(C\) such that \(2p_1 + 2p_2 + 2p_3 \sim 2D\). Indeed, we check that \(2p_2 + 2p_2 + 2p_3 \sim K_C - 2p_0^+ = K_C - (K_C - 2D) = 2D\). If \(2\cdot \ell_0\) is the tangent cone to \(\Gamma\) at \(p_0\), the divisor \(2p_0^+ + 2p_1 + \dots + 2p_3\) is cut on \(\Gamma\) by a conic tangent to \(\ell_0\) to \(p_0^+\) as well as to \(\Gamma\) at the points \(p_1, p_2, p_3\). This time one considers the incidence variety \[\mathscr{Q}^{(1)}:= \left\{(\Gamma, C, p_0, \dots, p_3)\colon \begin{matrix} C \in |\mathcal{O}_{\mathbb{P}^2}(5)| \text{ has a cusp at p_0, }C \in |\mathcal{O}_{\mathbb{P}^2}(2)| \\ \text{such that }\Gamma \cdot C = 4p_0 + 2p_1 + 2p_2 + 2p_3 \end{matrix}\right\}.\] The existence of a birational isomorphism \(\mathcal{T}^{(1)} \approx (\mathscr{Q}^{(1)}/ \! \! /\mathrm{PGL}(3))/\mathfrak{S}_3\) describing \(\mathcal{T}^{(1)}\) as an open subset of \((\mathscr{Q}^{(1)}/ \! \! /\mathrm{PGL}(3))/\mathfrak{S}_3\), giving access to the Chow ring of \(\mathcal{T}^{(1)}\). By similar means as described for \(\widetilde{\mathcal{S}}_5^-\), the Chow ring \(\mathrm{CH}^\bullet(\mathcal{T}^{(1)})\) is generated by restrictions of tautological classes from \(\mathcal{S}_5^-\). The tautology of \([\mathcal{T}^{(1)}] \in \mathrm{CH}^\bullet(\mathcal{T}\mathcal{S}_5^-)\) follows by Porteus’ formula since \(\mathcal{T}^{(1)}\) is given by \(h^0(C, \eta(-p_0^+)) \ne 0\). ◻
We can now conclude the proof of the main result of this work.
Proof of Theorem 1. The main theorem follows by assembling together Corollary 2, Theorem 9, and Proposition 10 which, by Excision and push and pull, give the tautology of \(\mathrm{CH}^\bullet(\mathcal{S}_5^-)\). ◻