An ordinary Gushel–Mukai variety with a single isolated node is the intersection of the Grassmannian \(G(2,5)\) with a nodal quadric and a linear space. We consider such intersections in dimension three, four and five.
We describe a flop between the blowup of such a variety and a quadric fibration over \({\mathbb{P}}^2\): at the level of derived categories, this flop establishes an equivalence between the categorical resolution of the
Kuznetsov component of the Gushel–Mukai variety and the derived category of modules on the even part of the Clifford algebra of the quadric fibration. As a first application, we extend a result of Kuznetsov and Perry to the nodal case, and we describe a
subfamily of rational, nodal Gushel–Mukai fourfolds whose Kuznetsov components admit a categorical resolution of singularities by an actual \(K3\) surface of degree two without a Brauer twist. This produces evidence for a
version of Kuznetsov’s rationality conjecture. We also describe the relation with Verra threefolds and fourfolds at the birational and categorical level. In particular, in the three-dimensional case, we investigate alternative birational models by
hyperbolic equivalence and by Kuznetsov’s spinor modifications. We show that the categorical resolution of the Kuznetsov component of a 1-nodal Gushel-Mukai threefold determines its birational class, and we explicitly construct another birational model,
which is again a conic fibration over \({\mathbb{P}}^2\) branched over a sextic, with the same Kuznetsov component up to autoequivalences.
For \(2\leq n\leq 5\), a smooth, ordinary Gushel–Mukai (GM) variety of dimension \(n\) is defined as a transverse intersection: \[X_n = G(2, 5)\cap Q\cap
{\mathbb{P}}^{n+4}\] where \(Q\) is a smooth quadric hypersurface in \({\mathbb{P}}^9\). This notion emerged in the classification of Fano varieties: in fact, it has been proven by
Mukai [1]–[3], up to a later result by Mella [4], that smooth Fano varieties of coindex 3, degree 10, and Picard number 1 can only occur in dimension \(3\leq
n\leq 6\), and are either ordinary GM varieties as above, or double covers of linear sections of \(G(2, 5)\) branched over ordinary GM varieties (called special GM varieties). These varieties present
remarkable features, and have been object of thorough study from several perspectives, including Hodge theory, birational geometry and derived categories, especially because of deep links with EPW varieties and hyperkähler manifolds. To any GM variety one
canonically associates a sextic hypersurface in \({\mathbb{P}}^5\), known as an Eisenbud–Popescu–Walter (EPW) sextic, together with its double cover, which is a hyperkähler fourfold called a double EPW sextic. The
pair consisting of a GM variety and its associated double EPW sextic exhibits many parallels with the relationship between cubic fourfolds and their Fano varieties of lines (which are again hyperkähler) and such pairs provide one of the few known sources
of Fano–hyperkähler correspondences in higher dimensions. The Hodge structure of GM varieties is also quite interesting (respectively, [5]–[8]):
In particular, in even dimensions, the middle Hodge structure of a GM variety coincides, up to a Tate twist, with the second cohomology of its double EPW sextic, and this identification, together with the Verbitsky–Torelli theorem for hyperkähler
fourfolds [9], yields a complete description of the period maps for GM varieties in dimensions 4 and 6. In odd dimensions the intermediate
Jacobian of a GM variety is similarly determined by the geometry of the associated EPW sextic, underlining the central role of EPW constructions in understanding the Hodge theory and moduli of GM varieties.
The notion of a Fano variety of \(K3\) type is expected to have a categorical counterpart at the level of derived categories in terms of a semiorthogonal decomposition containing a \(K3\)
category (i.e. its Serre functor is a shift by two and its Hochschild cohomology is the same as an actual \(K3\) surface). Indeed, in both the cubic fourfold and the GM fourfold cases, the bounded derived category of
coherent sheaves admits a semiorthogonal decomposition whose nontrivial component is a \(K3\) category. More precisely, for a smooth cubic fourfold \(X \subset \mathbb{P}^5\) one has a
semiorthogonal decomposition \[\mathrm{D}^b(Y)
=
\langle
\mathcal{K}u(Y),\,
\mathcal{O}_Y,\,
\mathcal{O}_Y(1),\,
\mathcal{O}_Y(2)
\rangle,\] where the Kuznetsov component \(\mathcal{K}u(Y)\) is a \(K3\) category [10]. Similarly, for a smooth ordinary GM fourfold \(Y\), there is a semiorthogonal decomposition \[\mathrm{D}^b(X)
=
\langle
\mathcal{K}u(X),\,
\mathcal{O}_X,\,
\mathcal{U}_X^\vee,\,
\mathcal{O}_X(1),\,
\mathcal{U}_X^\vee(1)
\rangle,\] where \(\mathcal{U}_X\) denotes the restriction of the tautological bundle from the Grassmannian, and the Kuznetsov component \(\mathcal{K}u(X)\) is again a \(K3\) category [11].
In both cases, moduli space of Bridgeland-stable objects on the Kuznetsov components has been constructed, and proven to be isomorphic to hyperkähler varieties, providing Fano-hyperkähler pairs in a completely categorical way [12]–[15].
While the Kuznetsov component of a general smooth cubic fourfold is expected to be non-geometric (i.e. not equivalent to the derived category of a \(K3\) surface), the corresponding component of the resolution of
singularities of a nodal cubic fourfold is geometric: inspired by the works [10], [16], we give a description of a “natural” Kuznetsov component of a resolution of a nodal ordinary GM fourfold and we describe a similar picture for threefolds and fivefolds.
Let us start by briefly reviewing what is known for cubic fourfolds.
Consider a cubic hypersurface in \(Y\subset{\mathbb{P}}^5\) with a single node \(y\in Y\), and its blowup \(\pi: \widetilde{Y}\longrightarrow Y\) in the
nodal point. Call \(E\) the exceptional divisor, and \(H\) the pullback of the hyperplane class of \({\mathbb{P}}^5\). Then, by applying Kuznetsov’s result
on categorical resolutions of singularities [17], there exists an admissible subcategory \(\widetilde{\mathcal{D}}\subset D^b(\widetilde{Y})\) such that one has a semiorthogonal decomposition \[\label{eq:cat95resolution95cubic954fold} D^b(\widetilde{Y}) = \langle {\mathcal{O}}_E(2E), {\mathcal{O}}_E(E), \widetilde{\mathcal{D}}\rangle,\tag{1}\] and such that the functors \(\pi_*:
\widetilde{\mathcal{D}}\longrightarrow D^b(Y)\) and \(\pi^*: {\mathcal{D}}^{\,\text{perf}}(Y)\longrightarrow{\mathcal{D}}\) are an adjoint pair (respectively right and left adjoint). However, this decomposition can
be refined to the following [10]:
In light of Equation 1 , \(\widetilde{{\mathcal{K}}u}(Y)\) can be interpreted as the categorical resolution of the Kuznetsov component. One can prove that \(\widetilde{{\mathcal{K}}u}(Y)\) is equivalent to the derived category of a \(K3\) surface \(Z\subset{\mathbb{P}}^4\) with the following argument [10]:
First, \(\widetilde{Y}\) is a divisor in the blowup \(\widetilde{\mathbb{P}}^5 := \operatorname{Bl}_y{\mathbb{P}}^5\), where the latter has a second contraction \(q:\widetilde{\mathbb{P}}^5\longrightarrow{\mathbb{P}}^4\) which is the projectivization of a vector bundle of rank two.
Call \(h\) the pullback of the hyperplane class from \({\mathbb{P}}^4\): then, as divisor classes in \(\widetilde{\mathbb{P}}^5\), one sees that \(\widetilde{Y} = 3H-2E = 2h+H\), i.e. the induced morphism \(\widetilde{Y}\longrightarrow{\mathbb{P}}^4\) is a blowup of \({\mathbb{P}}^4\) in the \(K3\) surface \(Z\) given by the vanishing of a section of \(q_*{\mathcal{O}}(2h+H)\simeq {\mathcal{O}}(2h)\oplus{\mathcal{O}}(3h)\). Let us call \(D\) the exceptional divisor, \(\bar q:= q|_E\) and \(j\) the embedding of \(D\) in \(\widetilde{Y}\).
By Orlov’s blowup formula [18], one has a semiorthogonal decomposition:
\[\label{eq:sod95resolution95nodal95cubic95right} D^b(\widetilde{Y}) = \langle j_*\bar q^*D^b(Z)\otimes{\mathcal{O}}(D), {\mathcal{O}}, {\mathcal{O}}(h),
{\mathcal{O}}(2h), {\mathcal{O}}(3h), {\mathcal{O}}(4h) \rangle.\tag{3}\]
Finally, comparing the decompositions 2 and 3 , one can prove that \(\widetilde{{\mathcal{K}}u}(Y) \simeq
D^b(Z)\) by a sequence of mutations of the exceptional objects generating their semiorthogonal complements.
We proceed with the discussion of nodal Gushel-Mukai \(n\)-folds.
Let \(X_n\) be an ordinary GM \(n\)-fold with a single node \(p\in X_n\). Then, the blowup \(\widetilde{X}_n:=\operatorname{Bl}_p X_n\) also admit a “nice” semiorthogonal decompositions in the spirit of 2 (see Proposition 11): \[D^b(\widetilde{X}_n) = \langle {\mathcal{O}}_E((n-2)E), \dots, {\mathcal{O}}_E(E),
\widetilde{{\mathcal{K}}u}(X_n), {\mathcal{O}}, {\mathcal{U}}^\vee, \dots, {\mathcal{O}}((n-3)H), {\mathcal{U}}^\vee((n-3)H)\rangle. \rangle\] The first goal of this paper is to give a geometric interpretation of \(\widetilde{{\mathcal{K}}u}(X_n)\).
Theorem 1. (Theorem 4). Let \(X_n(3\leq n\leq 5)\) be an
ordinary Gushel-Mukai \(n\)-fold with a single node \(p\in X_n\). Then the categorical resolution\(\widetilde{{\mathcal{K}}u}(X_n)\) is given by
\[\widetilde{{\mathcal{K}}u}(X_n)\simeq D^b(\mathbb{P}^2,\mathcal{C}_n),\] where \(D^b(\mathbb{P}^2,\mathcal{C}_n)\) is the derived category of coherent sheaves of modules on the even part
of the Clifford algebra (see 2.4 in [19]), associated to a quadric
fibration \(R_n\) over \(\mathbb{P}^2\) of relative dimension \(n-2\). In particular, \(\widetilde{{\mathcal{K}}u}(X_4)\simeq
D^b(\mathbb{P}^2,\mathcal{C}_4)\simeq D^b(S,\alpha)\), where \(S\) is a degree two \(K3\) surface and \(\alpha\in\mathrm{Br}(S)\) is a Brauer
class.
The natural strategy to give a simple geometric interpretation to \(\widetilde{{\mathcal{K}}u}(X_n)\), would be to adapt the approach of [10], described above for cubic fourfolds, and attempt to generalize it to \(X_n\). However, as we show in Section 2, the second contraction of
\(\widetilde{X}_n\) is not a blowup, but a small contraction to a singular \(n\)-fold \(T_n\subset {\mathbb{P}}^{n+3}\). Hence, there is no obvious, natural
semiorthogonal decomposition associated to such a contraction, and we have to rely on a more complex geometric picture.
In fact, a natural semiorthogonal decomposition can be found up to a birational transformation: by a modification of a construction from [20], we show that \(\widetilde{X}_n\) is birational to a quadric fibration \(R_n\) over \({\mathbb{P}}^2\). This map can be
obtained by resolving the projection \(G(2, 5)\dashrightarrow {\mathbb{P}}^8\) from the node, and blowing up the image of \(\widetilde{X}_n\) in a singular divisor (see [21] for a description of this construction in dimension three, and [22] for its four-dimensional analogue). Such birational map is a relative Atiyah flop, and it induces a derived equivalence \(D^b(\widetilde{X}_n)\simeq D^b(R_n)\). As a quadric fibration, \(R_n\) admits a rather simple semiorthogonal decomposition [19]: \[D^b(R_n) = \langle D^b({\mathbb{P}}^2, \mathcal{C}_n),
q^*D^b({\mathbb{P}}^2), \dots, q^*D^b({\mathbb{P}}^2)\otimes{\mathcal{O}}((n-3)h)\rangle\] where \(h\) is the relative hyperplane class, \(q:R_n\longrightarrow{\mathbb{P}}^2\) is the
quadric bundle map, and \(D^b({\mathbb{P}}^2, \mathcal{C}_n)\) is the derived category of modules on the even part of the Clifford algebra associated to \(R_n\). By comparing the latter
decomposition with Equation ?? , and applying a sequence of mutations, we prove that \(\widetilde{{\mathcal{K}}u}(X_n)\simeq D^b({\mathbb{P}}^2, \mathcal{C}_n)\).
Verra threefolds and fourfolds are Fano varieties whose double quadric fibration structure produces a rich geometry. In particular, a Verra threefold is a \((2,2)\)-section of \({\mathbb{P}}^2\)[23], and a Verra fourfold is a double cover of \({\mathbb{P}}^2\times{\mathbb{P}}^2\) branched in a Verra threefold [23], [24]. As a consequence of Theorem 1, and the flop \(\widetilde{X}_n\dashrightarrow R_n\) mentioned above, we describe the relationship with Verra threefolds and fourfolds.
If \(n=3\), then \(\widetilde{X_3}\) is known to be birationally equivalent to a Verra threefold \(V_3\)[25]. The latter has a natural semiorthogonal decomposition whose non-trivial component is also equivalent to the derived category
\(D^b(\mathbb{P}^2,\mathcal{C}_3')\) of modules over some Clifford algebra \(\mathcal{C}_3'\) on \(\mathbb{P}^2\), denoted by \({\mathcal{K}}u(V_3)\). Given a one-nodal GM threefold \(X_3\), and the conic fibration \(R_3\) birational to it, we prove that there is a Verra threefold \(V_3\) satisfying the following condition: there is a third quadric fibration \(W\) over \({\mathbb{P}}^2\) such that both \(X_3\) and \(R_3\) are obtained by \(W\) via the operation of hyperbolic reduction (see Section 4.1 and the sources
therein for more about hyperbolic reductions). This relation is often called hyperbolic equivalence. Then, by a combination of Theorem 4 and a result of [26], we conclude that \({\mathcal{K}}u(V_3)\simeq\widetilde{{\mathcal{K}}u}(X_3)\).
If \(n=4\), then \(\widetilde{X_4}\) is birationally equivalent to a Verra fourfold \(V_4\), whose nontrivial semiorthogonal component \({\mathcal{K}}u(V_4)\) is also a derived category of a twisted \(K3\) surface of degree two. In this case, the fact that the Kuznetsov components are equivalent can be deduced by Theorem 4, once we apply [22] to obtain the hyperbolic equivalence of \(R_4\) and \(V_4\).
We summarize the discussions above into the following proposition:
Proposition 1. (Corollary 3). Let \(X_n(3\leq n\leq 4)\) be a one-nodal Gushel-Mukai \(n\)-fold, then the categorical resolution \(\widetilde{{\mathcal{K}}u}(X_n)\) of the Kuznetsov components constructed in Theorem 1 is equivalent to Kuznetsov component \({\mathcal{K}}u(V_n)\) of some Verra \(n\)-fold.
Kuznetsov and Perry prove that for \(5\leq n\leq 6\), for a smooth GM \(n\)-fold, \(X_n\), there is a smooth GM \(n-2\)-fold \(X_{n-2}\) such that \({\mathcal{K}}u(X_n) \simeq {\mathcal{K}}u(X_{n-2})\). We extend this result for some nodal GM varieties, by replacing the
Kuznetsov component with its categorical resolution. More precisely, we have:
Choose two general hyperplane sections \(s_1, s_2\) of \(G(2, 5)\) passing through the node \(p\). Then, in the space of quadric hypersurfaces in
\({\mathbb{P}}^9\) which are nodal in \(p\), there is a codimension two subspace \(\EuScript{H}\) such that for the general \(\EuScript{Q}\in\EuScript{H}\), the Gushel–Mukai fivefold \(X_5:= G(2, 5)\cap\EuScript{Q}\) enjoys the following property: there is a nodal Gushel–Mukai threefold \(X_3
= {\mathbb{V}}(s_1, s_2)\cap X_5\) such that \(\widetilde{{\mathcal{K}}u}(X_3)\simeq \widetilde{{\mathcal{K}}u}(X_5)\).
Choose three general hyperplane sections \(s_1, s_2, s_3\) of \(G(2, 5)\) passing through \(p\). Then, in the space of quadrics in \({\mathbb{P}}^9\) nodal in \(p\), there is a codimension two subspace \(\EuScript{H}\) such that for the general \(\EuScript{Q}\in\EuScript{H}\), the Gushel–Mukai fourfold \(X_4:= G(2, 5)\cap\EuScript{Q}\cap{\mathbb{V}}(s_1)\) enjoys the following property: there is a smooth \(K3\) surface of degree two \(\widetilde{X}_2\), which is the blowup of \(X_2 = {\mathbb{V}}(s_2, s_3)\cap X_4\) in \(p\), such
that \(D^b(\widetilde{X}_2)\simeq \widetilde{{\mathcal{K}}u}(X_4)\).
While the general one-nodal GM fivefold is rational, this is not true for fourfolds. However, the fourfolds we consider in point \((2)\) of Theorem 2 are indeed rational, since they are birational to rational quadric fibrations, and this explains the vanishing of the Brauer class. Hence, Theorem 2 characterizes the categorical resolution of the Kuznetsov component as “geometric”, i.e. equivalent to the derived category of coherent sheaves on a \(K3\) surface. This
result confirms a GM analogue of the Kuznetsov’s rationality conjecture by allowing singularities, which predicts that a singular Gushel-Mukai fourfold is rational if and only if the Kuznetsov component admits a categorical resolution of singularities by
an actual \(K3\) surface. We also note that Theorem 1 and Theorem 2(2) are in the same spirit of the results obtained in [27], where
the authors consider singular quartic double fivefolds.
Categorical Torelli theorem for nodal Gushel–Mukai threefolds↩︎
As a further application, we prove a (birational) categorical Torelli theorem for the 1-nodal Gushel-Mukai threefold \(X_3\). In [28], the authors show that the Kuznetsov component \({\mathcal{K}}u(X)\) of a general smooth Gushel-Mukai threefold \(X\) characterizes the birational class of \(X\). In [29], the authors
show that the Kuznetsov component \({\mathcal{K}}u(X')\) of a 1-nodal non-factorial Gushel-Mukai threefold \(X'\) determines its birational class, and it is clear that its
categorical absorption(in the sense of [30]) \(\widetilde{A}_{X'}\subset{\mathcal{K}}u(X')\) determines the birational class as well. For 1-nodal factorial Gushel-Mukai threefold \(X_3\), we prove the following:
Theorem 3. (Theorem 8). Let \(X_3\) be a 1-nodal factorial Gushel-Mukai threefold, then the categorical
resolution \(\widetilde{{\mathcal{K}}u}(X_3)\) determines its birational class among 1-nodal factorial Gushel-Mukai threefolds.
In a recent work [31], Kuznetsov developed a new tool called spinor modification, to relate different conic
bundles on the same base, without the need of an explicit hyperbolic equivalence (and hence without relying on the construction of a higher-dimensional quadric bundle which is related to both of them by the operation of quadric reduction). The even
Clifford algebras of conic bundles related by a spinor modification are Morita equivalent. While it is known that hyperbolic-equivalent conic bundles are related by a spinor modification [31], it is not clear whether the converse holds: spinor modification induces birational equivalence, and conjecturally hyperbolic equivalence [31]. Therefore, it would be interesting to construct an explicit, abstract spinor bundle on \(R_3\) which
realizes \(V_3\) as a spinor modification, thus producing new evidence for the conjecture, but this taksed proved to be elusive. However, we define an abstract spinor bundle on \(R_3\) which
induces a spinor modification to a conic fibration in the projectivization of \({\mathcal{O}}\oplus{\mathcal{O}}(-1)\oplus{\mathcal{O}}(-2)\), thus finding a new birational model for the conic fibrations \(\widetilde{X}_3, R_3, V_3\), which shares the same Kuznetsov component up to autoequivalence. Applying a similar technique to \(V_3\) produces a “trivial” spinor modification, i.e. an
isomorphism: in fact, we show that the associated abstract spinor bundle is canonical, in the sense of [31].
This behavior is similar to what happens for resolutions of one-nodal Fano threefolds of genus 5 [31]
The authors would like to express their gratitude towards Grzegorz and MichałKapustka for introducing them to this problem, and to Arend Bayer, Alexander
Kuznetsov, Laurent Manivel for their valuable comments on the first draft. Moreover, MR would like to thank Ignacio Barros, Soheyla Feyzbakhsh, Giovanni Mongardi and Ying Xie for helpful and inspiring suggestions. SZ thanks Zhiyu Liu, Xun Lin, Fei Peng,
and Zheng Zhang for their useful discussions. Part of the work was carried out when the authors visited IBS-CGP, IMS-Shanghai Tech University, the Jagiellonian University, IMPAN, and Soochow University. We are grateful for their excellent hospitality and
support. KG was supported by 2024/53/B/ST1/00161. MR was supported by Fonds voor Wetenschappelijk Onderzoek – Vlaanderen (FWO, Research Foundation – Flanders) – G0D9323N, by the European Research Council (ERC) grant agreement No. 817762, and by the Polish
National Science Center project number 2024/55/B/ST1/02409. SZ is supported by SYSU Starting Grant No. 34000-12256019. The research was supported by the program Excellence Initiative at the Jagiellonian University in Krakow (ID.UJ).
We shall work over the field of complex numbers. Given \(m\) equations \(f_1, \dots, f_m\), with the notation \({\mathbb{V}}(f_1, \dots, f_m)\) we denote
the intersection of their vanishing loci. We adopt the same notation when \(f_1, \dots, f_m\) are sections of vector bundle. We use the symbol “\(\simeq\)” for isomorphisms, and “\(\sim\)” for birational equivalences. By \(D^b(X)\), we denote the bounded derived category of coherent sheaves of a variety \(X\).
Let us fix \(\EuScript{V}\simeq {\mathbb{C}}^5\), and a decomposition \(\EuScript{V}= \EuScript{P}\oplus \EuScript{W}\) where \(\EuScript{P}\) has
dimension two. This will induce a decomposition: \[\wedge^2\EuScript{V}= \wedge^2\EuScript{P}\oplus \EuScript{P}\otimes \EuScript{W}\oplus \wedge^2 \EuScript{W}.\] We call \(\operatorname{Gr}:=G(2, \EuScript{V})\) the Grassmannian parametrizing two-dimensional subspaces of \(\EuScript{V}\). This is a six-dimensional smooth projective variety in \({\mathbb{P}}(\wedge^2 \EuScript{V})\). Define \(H\) to be the Plücker hyperplane class. We call \(Q_p\) a quadric hypersurface in \({\mathbb{P}}(\wedge^2 \EuScript{V})\) with a single node \(p = {\mathbb{P}}(\wedge^2(\EuScript{P}))\in \operatorname{Gr}\). In the following, we shall denote by \(X_n\) the intersection in \({\mathbb{P}}(\wedge^2 \EuScript{V})\) of \(\mathop{\mathrm{Gr}}\) with \(Q_p\) and a linear space of
dimension nine, eight or seven. This is a one-nodal Gushel–Mukai variety of dimension, respectively, five, four or three. Let us call \(n\) the dimension of such variety: then, it is realized inside \(\operatorname{Gr}\), as the zero locus of a section of \({\mathcal{O}}(2H)\oplus\EuScript{K}\otimes{\mathcal{O}}(H)\) where \(\EuScript{K}\) is a vector space of
dimension \(5-n\).
Consider the projection \[\begin{tikzcd} \pi_p: {\mathbb{P}}(\wedge^2 \EuScript{V})\ar[dashed]{r} & {\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W}) \end{tikzcd}\] from \(p\). The image \(\EuScript{Q}:=\pi_p(Q_p)\) is a smooth, seven-dimensional quadric hypersurface in \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus
\wedge^2\EuScript{W})\). On the other hand, if we restrict \(\pi_p\) to \(\operatorname{Gr}\), the resulting map contracts all lines in \(\operatorname{Gr}\) which pass through \(p\). Let us identify the node with a decomposable two-vector \(p = [p_1\wedge p_2]\in {\mathbb{P}}(\wedge^2
\EuScript{V})\): the contracted locus of \(\pi_p |_{\operatorname{Gr}}\) is given by:
Summing all up, the image of the contracted locus of \(\pi_p|_{X_n}\) is the cone over the intersection of \({\mathbb{P}}(\EuScript{P})\times{\mathbb{P}}(\EuScript{W})\) with a smooth
quadric and a linear space of dimension \(n\) in \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})\). Let us call such locus \(\Gamma\). Since the degree of
\({\mathbb{P}}(\EuScript{P})\times{\mathbb{P}}(\EuScript{W})\subset{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})\) is three, we conclude that \(\Gamma\) is a smooth \(n-3\)-fold of degree six (six distinct points for \(n=3\)).
The following lemma is a consequence of [20]. For the reader’s convenience, and to ensure uniformity of notation with the rest of the
paper, we provide a proof adapted to our setting.
Lemma 2. The image of \(\pi_p|_{\operatorname{Gr}}\) in \({\mathbb{P}}(\EuScript{W}\otimes\EuScript{P}\oplus\wedge^2\EuScript{W})\) is the intersection of two Plücker quadrics,
and it contains \({\mathbb{P}}(\EuScript{W}\otimes\EuScript{P})\).
Proof. let us choose coordinates \(x_1, \dots, x_5\) for \(\EuScript{V}\), and \(x_{12}, \dots, x_{45}\) for \(\wedge^2
\EuScript{V}\), such that: \[p = [1:0\dots 0] = {\mathbb{V}}(x_{13}, \dots, x_{45}).\] Hence, we have: \[\begin{align} \EuScript{P}& = {\mathbb{V}}(x_3, x_4, x_5) \subset
\EuScript{V}\\ \EuScript{W}& = {\mathbb{V}}(x_1, x_2) \subset \EuScript{V}. \end{align}\] The Grassmannian \(\operatorname{Gr}\) is the zero locus of the following Plücker quadrics: \[\begin{align} & P_1: \quad x_{12} x_{34} - x_{13} x_{24} + x_{14} x_{23} = 0 \\ & P_2: \quad x_{12} x_{35} - x_{13} x_{25} + x_{15} x_{23} = 0 \\ & P_3: \quad x_{12} x_{45} - x_{14} x_{25} + x_{15} x_{24} = 0 \\ &
Q_1: \quad x_{13} x_{45} - x_{14} x_{35} + x_{15} x_{34} = 0 \\ & Q_2: \quad x_{23} x_{45} - x_{24} x_{35} + x_{25} x_{34} = 0 \end{align}\] The image of the projection from \(p\) is thus the zero locus of \(Q_1\), \(Q_2\). In fact, we can use \(P_1\), \(P_2\), \(P_3\) to eliminate \(x_{12}\) and obtain three cubic relations which identically vanish in the zero locus of \(Q_1\), \(Q_2\). The proof is completed once we observe that the latter
contains the linear space given by the conditions \(x_{34}=x_{35}=x_{45}=0\), which is precisely \({\mathbb{P}}(\EuScript{W}\otimes\EuScript{P})\). ◻
One immediately has the following:
Corollary 1. The variety \(T_n:=\operatorname{im}(\pi_p|_{X_n})\subset{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W})\) is the intersection of the smooth quadric \(\EuScript{Q}\) with \(Q_1\), \(Q_2\) and a linear space of codimension \(5-n\). Moreover, the singular locus of \(T_n\) is \(\Gamma\).
Let us now blow up \(p\) and investigate the second contraction of the resulting variety, i.e. the resolution of \(\pi_p|_{X_n}\). One has \(\mathop{\mathrm{Bl}}_p {\mathbb{P}}(\wedge^2 \EuScript{V})\simeq{\mathbb{P}}({\mathcal{O}}\oplus{\mathcal{O}}(-h)\longrightarrow{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W}))\), where \(h\) is the hyperplane class of \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W})\). Restricting the blowup to \(X_n\), we find the
following picture: \[\begin{tikzcd}[row sep = huge, column sep = huge] E\ar[hook]{r}\ar{d}{\bar\phi} & \widetilde{X}_n \ar[hookleftarrow]{r}\ar{d}{\phi}\ar{dr}{\pi} & S_X\ar{dr}{\alpha} & \\ p\ar[hook]{r} & X_n
\ar[dashed, swap]{r}{\pi_p|_{X_n}} & T_n\ar[hookleftarrow]{r} & \Gamma \end{tikzcd}\] where \(\pi\) is the resolution of the projection, \(E\) is the exceptional divisor of
the blowup (which is a smooth quadric), and \(T_n\) is the image of the projection \(\pi_p|_{X_n}\). The restriction of \(\pi\) to the contracted locus
yields a \({\mathbb{P}}^1\)-bundle \(\alpha: S_X\longrightarrow\Gamma\).
Lemma 3. One has the following linear relations: \[\begin{align} E &= H-h\\ K_{\widetilde{X}_n} & = -(n-2)h \end{align}\]
Proof. By equating the expressions of the canonical class of \(\mathop{\mathrm{Bl}}_p {\mathbb{P}}(\wedge^2 \EuScript{V})\) as a blowup and as a projective bundle, one can easily see that \(E = H-h\). Moreover, by the normal bundle sequence associated to the embedding of \(\widetilde{X}_n\) in the blowup of \(\operatorname{Gr}\), one can compute the
canonical class of \(\widetilde{X}_n\). To do this, let us first observe that the blowup \(\widetilde{X}_5\) sits as a divisor inside the blowup \(\widetilde{\operatorname{Gr}}\). Hence, as divisor classes, one has \(\widetilde{X}_5 = 2H-2E = 2h\). Therefore, the normal bundle of \(\widetilde{X}_n\) in
\(\mathop{\mathrm{Bl}}_p\operatorname{Gr}\) will be \({\mathcal{O}}(2h)\oplus\EuScript{K}\otimes{\mathcal{O}}(h)\). Then we conclude by adjunction, once we note that \(\omega_{\mathop{\mathrm{Bl}}_p\operatorname{Gr}}\simeq{\mathcal{O}}(-5H+5E) \simeq{\mathcal{O}}(-5h)\). ◻
Let us now consider the blowup of \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W})\) with center \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})\). We obtain the
following diagram: \[\begin{tikzcd}[row sep = huge, column sep = huge] {\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}) \times{\mathbb{P}}(\wedge^2\EuScript{W})\ar[hook]{r}\ar{d} &
\mathop{\mathrm{Bl}}_{{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})}{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W})\ar{d}{\rho}\ar{dr} & \\ {\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})\ar[hook]{r} &
{\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus \wedge^2\EuScript{W}) & {\mathbb{P}}( \wedge^2\EuScript{W}) \end{tikzcd}\] where the diagonal arrow comes from the isomorphism of the blowup with the projective bundle \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\otimes{\mathcal{O}}\oplus{\mathcal{O}}(-\zeta)\longrightarrow{\mathbb{P}}(\wedge^2\EuScript{W}))\). Here \(\zeta\) denotes the hyperplane class of
\({\mathbb{P}}(\wedge^2\EuScript{W})\).
The following result is formulated in [20] in greater generality. However, let us rephrase it specifically for our picture:
Proposition 4. [20]
The strict transform of \(Q_1\cap Q_2\) is isomorphic to the projective bundle \({\mathbb{P}}(\EuScript{P}\otimes\Omega^1_{{\mathbb{P}}(\EuScript{W})}(\zeta)\oplus
{\mathcal{O}}(-\zeta))\longrightarrow{\mathbb{P}}(\EuScript{W})\). Moreover, one has \(\zeta = 2h-H\).
Restricting to a linear section, we find:
Lemma 5. Call \(Z\subset Q_1\cap Q_2\) the zero locus of a general section of \(\EuScript{K}\otimes{\mathcal{O}}(h)\). Then the strict transform \(\widetilde{Z}\) of \(Z\) is a projective bundle of rank \(n-1\) over \({\mathbb{P}}(\wedge^2\EuScript{P})\).
Proof. For \(n=5\) this is simply Proposition 4, so let’s assume \(n = 3,
4\). By applying [20], we see that \(\widetilde{Z}\longrightarrow{\mathbb{P}}(\wedge^2\EuScript{W})\) is a fibration with general fiber isomorphic to \({\mathbb{P}}^{n-1}\). Such fibration has degenerate fibers on the degeneracy
locus of a general morphism:
By dimension count, we see that the degeneracy locus is empty for both \(n=3\) and \(n=4\): thus, \(\ker(\nu)\) is a vector bundle of rank \(n\), and \(\widetilde{Z}\simeq {\mathbb{P}}(\ker(\nu))\). ◻
By further restricting \(\rho\), we deduce the following:
Lemma 6. Let us call \(R_n\) the strict transform of \(T_n\) with respect to \(\rho\). Then, there is a diagram: \[\begin{tikzcd} & S_R \ar[hook]{r}\ar[swap]{dl}{\beta} & R_n \ar[hookleftarrow]{r}\ar{dl}{\rho}\ar[swap]{rd}{\psi} & \psi^{-1}(\Delta)\ar{dr} & \\ \Gamma \ar[hook]{r} & T_n & &
{\mathbb{P}}(\wedge^2\EuScript{W}) \ar[hookleftarrow]{r} & \Delta \end{tikzcd}\] where:
\(\psi: R_n\longrightarrow{\mathbb{P}}(\wedge^2\EuScript{W})\) is a quadric fibration of relative dimension \(n-2\), and its discriminant divisor \(\Delta\) is a smooth curve of degree six
The restriction of \(\rho\) to \(R_n\) is a small contraction, which contracts a \({\mathbb{P}}^1\)-bundle \(S_R\) to
\(\Gamma\subset T_n\).
The canonical bundle of \(R_n\) is \(\omega_{R_n} \simeq {\mathcal{O}}(-(n-2)h)\)
Proof. Since \({\mathcal{O}}(h)\) is a relative hyperplane bundle with respect to the projective bundle structure \(\widetilde{Z}\longrightarrow{\mathbb{P}}(\wedge^2\EuScript{W})\), it is clear that \(R_n\) is a quadric fibration of relative dimension \(n-2\). The degree of
the discriminant is given by the degree of \(\det(\ker(\nu^\vee))^{\otimes 2} = {\mathcal{O}}(6\zeta)\). Smoothness follows by dimensional reasons.
To prove the second point, let us observe that the strict transform is a blowup of \(T_n\) in a divisor \(D\) given by intersecting \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W})\) with a general section of \(\EuScript{K}\otimes{\mathcal{O}}(h)\oplus{\mathcal{O}}(2h)\). The singular locus of \(T_n\), which we called \(\Gamma\), is contained in \(D\), and its preimage through the strict transform is the projectivization of the normal bundle \({\mathcal{N}}_{\Gamma | D}\). The latter is a rank two vector bundle.
To address the last item, recall that \(R_n\) is the zero locus of a section of \({\mathcal{O}}(2h)\) in \(\widetilde{Z}\), and the latter is the
projectivization of \(\ker(\nu)\) over \({\mathbb{P}}(\wedge^2\EuScript{W})\). Then, by combining Equation 4 with the relative tangent bundle sequence of
\(\widetilde{Z}\longrightarrow{\mathbb{P}}(\wedge^2\EuScript{W})\), we obtain \(\omega_{\widetilde{Z}} \simeq {\mathcal{O}}(-nh)\), and we conclude by adjunction. ◻
Summing all up, we obtain a birational map \(\mu:\widetilde{X}_n\dashrightarrow R_n\) with contracted loci given by \(S_X\) and \(S_R\) respectively. We
can resolve such map by blowing up \(S_X\) and \(S_R\). By Lemma 3 and
Lemma 6, we see that \(\mu\) is crepant: hence, by [32], both blowups have the same exceptional divisor, which we will denote by \(F\). Such divisor has a \({\mathbb{P}}^1\times{\mathbb{P}}^1\) fibration over
\(\Gamma\) given by the map \(\pi:=\alpha\circ\bar{p}=\beta\circ\bar{q}\). Such construction is a relative Atiyah flop, which is an example of a generalized Grassmann flop[33]–[38].
\[\label{eq95keqdiagram} \begin{tikzcd}[row sep=large, column sep = normal] & & F\ar[hook]{d}{\iota}\ar[swap]{lldd}{\bar p}\ar{rrdd}{\bar q} & &\\ & &
{\mathcal{X}}\ar[swap]{ld}{p}\ar{rd}{q} & &\\ S_X\ar[hook]{r}\ar[swap]{rrdd}{\alpha}& \widetilde{X}_n\ar[dashed]{rr}{\mu}\ar[swap]{dr} & & R_n\ar[hookleftarrow]{r}\ar{dl}{\rho} &S_R\ar{lldd}{\beta}\\
&&T_n\ar[hookleftarrow]{d}&&\\ &&\Gamma&& \end{tikzcd}\tag{5}\]
Remark 7. One can also see \(\mu\) as the restriction to \(\widetilde{X}_n\) of the birational map described in [20] where, in their notation, X\(=\operatorname{Gr}\). See, in particular, [20].
Remark 8. Let us briefly describe how to extend Diagram 5 to the case of nodal Gushel–Mukai surfaces. Let us call \(\widetilde{X}_2\) the blowup of a nodal \(K3\) surfsce of degree 10 in the node. Then, \(\widetilde{X}_2\) is a smooth \(K3\) surface [39]. The restriction of the morphism \(\widetilde{X}_3\longrightarrow T_3\) to a general hyperplane section is an isomorphism (the hyperplane in
\(T_3\) avoids \(\Gamma\)), and therefore \(\rho\) restricts to the blowup of a smooth divisor (another isomorphism). We get the following picture: \[\begin{tikzcd} E\ar[hook]{r}\ar{d} & \widetilde{X}_2\ar{d}\ar[dashed]{rr}{\mu}\ar[equals]{dr} & & R_2\ar{d}{2:1}\ar[equals]{dl}\ar[hookleftarrow]{r} & \Delta\ar[equals]{d}\\ \{p\}\ar[hook]{r} & X_2 & T_2 &
{\mathbb{P}}^2 \ar[hookleftarrow]{r} & \Delta \end{tikzcd}\] Summing all up, this diagram recovers the description of the second contraction of \(\widetilde{X}_2\) as a double cover of \({\mathbb{P}}^2\) branched in the sextic curve \(\Delta\). Therefore, \(\widetilde{X}_2\) is a smooth \(K3\) surface of degree 2
as expected.
Remark 9. Besides ordinary GM varieties, one can consider special ones, i.e. double covers of a linear section of \(G(2, \EuScript{V})\) branched in an ordinary GM variety. In fact, one can carry out a
geometric construction like the one we describved abovem, and lift it to double covers, realizing a similar flop picture. However, the at the level of derived categories, the picture becomes substantially more complicated, and it is currently unclear to
the authors if the approach of Section 3 can be suitably generalized to it. This will be object of further study.
3 The categorical resolution of the Kuznetsov component↩︎
The description of \(\mu\) as an Atiyah flop (Diagram 5 ) lifts to a derived equivalence \(D^b(\widetilde{X}_n)\simeq D^b(R_n)\)[40]. Let us give an explicit construction of the equivalence functor.
To this purpose, we briefly recall the notion of mutations of a semiorthogonal decomposition. While we summarize the minimal technical notions to keep the exposition self-contained, we refer to [41] for a thorough explanation of the subject. Given a semiorthogonal decompositon \({\mathcal{T}}\subset \langle{\mathcal{A}},
{\mathcal{B}}\rangle\) of a triangulated category \({\mathcal{T}}\), there are admissible subcategories \({\mathbb{L}}_{\mathcal{A}}{\mathcal{B}}\subset{\mathcal{T}}\) and \({\mathbb{R}}_{\mathcal{B}}{\mathcal{A}}\subset{\mathcal{T}}\) such that one has:
These operations on semiorthogonal decompositions are called mutations. While we will not need the explicit description of \({\mathbb{L}}_{\mathcal{A}}{\mathcal{B}}\) and \({\mathbb{R}}_{\mathcal{B}}{\mathcal{A}}\) in full generality, let us recall what happens in the case in which \({\mathcal{A}}\) and \({\mathcal{B}}\) are
generated by a single object. Set \({\mathcal{A}}= \langle A\rangle\) and \({\mathcal{B}}= \langle B\rangle\), where \(A\) and \(B\) are exceptional objects. Then, the corresponding mutations are also generated by a single exceptional object, which has the following expression:
\[\begin{align} {\mathbb{L}}_A B & = \operatorname{Cone}(A\otimes R^\bullet\operatorname{Hom}(A, B)\longrightarrow B) \\ {\mathbb{R}}_B A & = \operatorname{Cone}(A \longrightarrow R^\bullet\operatorname{Hom}(A,
B)^\vee\otimes B)[-1] \end{align}\]
Note that if \(R^\bullet\operatorname{Hom}(A, B)\) is concentrated in degree zero, then one has:
If \(A\otimes R^\bullet\operatorname{Hom}(A, B)\longrightarrow B\) is injective (surjective), then, up to shift, \({\mathbb{L}}_A B\) is quasi-isomorphic to its cokernel
(kernel)
If \(A \longrightarrow R^\bullet\operatorname{Hom}(A, B)^\vee \otimes B\) is injective (surjective), then, up to shift, \({\mathbb{R}}_B A\) is quasi-isomorphic to its cokernel
(kernel)
Moreover, if if \(R^\bullet\operatorname{Hom}(A, B)\) is concentrated in degree one and has dimension one, \({\mathbb{L}}_A B\) and \({\mathbb{R}}_B A\)
are quasi-isomorphic to the corresponding unique extension \(C\) fitting in a short exact sequence \[0\longrightarrow B\longrightarrow C\longrightarrow A\longrightarrow 0.\]
Proposition 10. There is an equivalence of triangulated categories: \[q^*D^b(R_n) \simeq \left[{\mathbb{R}}_{\iota_*\pi^*D^b(\Gamma)}p^*D^b(\widetilde{X}_n)\right]\otimes{\mathcal{O}}(-H)\]
Proof. By Orlov’s blowup formula [18] we have the following semiorthogonal decompositions:
Now recall that both \(S_X\) and \(S_R\) are \({\mathbb{P}}^1\)-bundles over \(\Gamma\): hence, by Orlov’s projective
bundle formula [18] we obtain the following:
where \(\xi\) can be chosen as any line bundle on \(F\) which restricts to \({\mathcal{O}}(1)\) on the fibers of \(\beta\). Then, we can recover the equivalence by a sequence of mutations, which will be outlined here.
Let us first apply the Serre functor to the first block of the first decomposition of Equation 6 . We obtain the following: \[D^b({\mathcal{X}}) = \langle
\iota_*\pi^*D^b(\Gamma)\otimes{\mathcal{O}}(F+ H), p^*D^b(\widetilde{X}_n), \iota_*\pi^*D^b(\Gamma)\otimes {\mathcal{O}}(F-K_{\mathcal{X}}) \rangle.\] Then, we mutate the central piece one step to the right, and we twist everything by \({\mathcal{O}}(-H)\)\[\begin{align} D^b({\mathcal{X}}) = & \langle \iota_*\pi^*D^b(\Gamma)\otimes{\mathcal{O}}(F), \iota_*\pi^*D^b(\Gamma)\otimes {\mathcal{O}}(F-H-K_{\mathcal{X}}), \\
&{\mathcal{O}}(-H)\otimes {\mathbb{R}}_{\iota_*\pi^*D^b(\Gamma)\otimes p^*\omega_{\widetilde{X}}^\vee}p^*D^b(\widetilde{X}_n) \rangle. \end{align}\] Observe that \(\iota_*\pi^*D^b(\Gamma)\), as an admissible
subcategory in the SOD above, is invariant with respect to tensor product with powers of \({\mathcal{O}}(h)\), hence, to conclude, we just need to show that \((-H-K_{\mathcal{X}})|_F = \xi\)
up to powers of \(h\). To this purpose, we just observe that \(K_{\mathcal{X}}= K_{\widetilde{X}} + F\), which, by Lemma 3, equals \((2-n)h+F\). By definition, \(F\) restricts to \(-H-\xi\) on \(F\), up to powers of \(h\), and this completes the proof. ◻
In the remainder of this section, we prove that the categorical resolution of the Kuznetsov component of \(X_n\) is equivalent to the derived category of modules on the even part of the Clifford algebra of \(R_n\).
The content of this first proposition is well-known to experts. Since the authors are not aware of any reference, let us give a self-contained proof.
Proposition 11. There is the following semiorthogonal decompositon: \[\label{eq:sod95resolved95gm95nfold} D^b(\widetilde{X}_n) = \langle {\mathcal{O}}_E((n-2)E),
\dots, {\mathcal{O}}_E(E), \widetilde{{\mathcal{K}}u}(X_n), {\mathcal{O}}, {\mathcal{U}}^\vee, \dots, {\mathcal{O}}((n-3)H), {\mathcal{U}}^\vee((n-3)H)\rangle.\qquad{(1)}\]
Proof. Let us illustrate the case \(n=5\), since the fourfold and threefold cases can be proved identically. Call \(\widetilde{\operatorname{Gr}}\) the blowup of \(\operatorname{Gr}\) in the node \(p\).
Let us first prove that \(\langle {\mathcal{O}}_{\widetilde{X}_5}, \phi^*{\mathcal{U}}^\vee, \dots, {\mathcal{O}}_{\widetilde{X}_5}(2H), \phi^*{\mathcal{U}}^\vee(2H)\rangle\) is an exceptional collection. This is equivalent
to showing that \({\mathcal{O}}_{\widetilde{X}_5}\) has a one dimensional space of sections and no higher cohomology, and that the following bundles are acyclic on \(\widetilde{X}_5\) for
\(1\leq m\leq 2\): \[{\mathcal{O}}(-mH){\mathcal{U}}^\vee(-mH){\mathcal{U}}^\vee(-(m+1)H)\mathop{\mathrm{Sym}}^2{\mathcal{U}}^\vee(-(m+1)H)\] To this purpose, let us begin by recalling
that, as divisor classes in \(\widetilde{\operatorname{Gr}}\), one has \(\widetilde{X}_5 = 2H-2E = 2h\). Hence, there is the Koszul sequence
To compute the cohomology of \({\mathcal{O}}_{\widetilde{X}_5}\) we first resolve it with Equation 7 and note that the middle term has \(h^0 =
1\) and no further cohomology. The first term is \({\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(-2h) = {\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(2E-2H)\), and can therefore be resolved by \[0\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(E-2H)\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(2E-2H)\longrightarrow{\mathcal{O}}_{{\mathbb{P}}^5}(-2)\longrightarrow 0.\] The last term has no
cohomology, while the first one is resolved by \[0\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(-2H)\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(E-2H)\longrightarrow{\mathcal{O}}_{{\mathbb{P}}^5}(-1)\longrightarrow 0,\] and therefore has
vanishing cohomology as well.
The cohomology of \({\mathcal{O}}_{\widetilde{X}_5}(-H)\) can be computed by tensoring Equation 7 by \({\mathcal{O}}(-H)\): the middle term is
clearly acyclic, and the first one is isomorphic to \({\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(-3H+2E)\). The latter, in light of Equation 8 , sits in a short exact
sequence: \[0\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(-3H+E)\longrightarrow{\mathcal{O}}_{\widetilde{\operatorname{Gr}}}(-3H+2E)\longrightarrow{\mathcal{O}}_{{\mathbb{P}}^5}(-2)\longrightarrow 0.\] The
last term has no cohomology and the first one, again by twisting Equation 8 , is resolved by \({\mathcal{O}}(-3H)\) and \({\mathcal{O}}_{{\mathbb{P}}^5}(-1)\), both acyclic. The cohomology of \({\mathcal{O}}_{\widetilde{X}_5}(-2H)\) can be computed in the same way, by iterating this approach. To prove that the
remaining bundles are acyclic, one adopts essentially the same technique with the additional caveat that \({\mathcal{U}}^\vee|_{{\mathbb{P}}^5}\simeq{\mathcal{O}}^{\oplus 2}\) and that \(\mathop{\mathrm{Sym}}^2{\mathcal{U}}^\vee|_{{\mathbb{P}}^5}\simeq{\mathcal{O}}^{\oplus 3}\). Additionally, one needs to show that \(H^\bullet(\mathop{\mathrm{Gr}}, {\mathcal{U}}^\vee(-kH)) =
H^\bullet(\mathop{\mathrm{Gr}}, \mathop{\mathrm{Sym}}^2{\mathcal{U}}^\vee(-kH)) = 0\) for \(1\leq 5\), and this can be checked by means of the Borel–Weil–Bott theorem: see [42] for a thorough introduction on the topic, and [43] for a simple
online calculator.
The next step is showing that \(\langle {\mathcal{O}}_E(3E), {\mathcal{O}}_E(2E), {\mathcal{O}}_E(E)\rangle\) is semiorthogonal. Call \(j: E\longrightarrow\widetilde{X}_5\) the embedding of
the exceptional quadric. One has: \[\begin{align} \operatorname{Hom}_{\widetilde{X}_5}^\bullet ( j_*{\mathcal{O}}_E, j_*{\mathcal{O}}(mE)) & \simeq \operatorname{Hom}_{E}^\bullet( j^* j_*{\mathcal{O}}_E, {\mathcal{O}}_E(mE))
\\ &\simeq \operatorname{Hom}_{E}^\bullet( j^* j_*{\mathcal{O}}_E, {\mathcal{O}}_E(-mh)). \end{align}\] Then, by [44], there is a distinguished triangle: \[{\mathcal{O}}_E(1)[1]\longrightarrow j^* j_*{\mathcal{O}}_E\longrightarrow{\mathcal{O}}_E \longrightarrow{\mathcal{O}}_E(1)[2]\] and we conclude by
applying the long exact sequence of \(\operatorname{Hom}_E^\bullet(-, {\mathcal{O}}_E(-mh))\).
The remainder of the proof consists of showing that for any element of the exceptional collection above (call it \({\mathcal{E}}\)) on \(X_5\), one has \(R^\bullet
\operatorname{Hom}_{\widetilde{X}_5}({\mathcal{E}}, {\mathcal{O}}_E(-kh)) = 0\) for \(1\leq k\leq 3\). One has: \[\begin{align} R^\bullet \operatorname{Hom}_{\widetilde{X}_5}({\mathcal{E}},
{\mathcal{O}}_E(-kh)) & \simeq H^\bullet(\widetilde{X}_5, {\mathcal{E}}^\vee\otimes{\mathcal{O}}_E(-kh)) \\ &\simeq H^\bullet(E, {\mathcal{O}}_E^{\oplus\operatorname{rk}{\mathcal{E}}}(-kh)) = 0. \end{align}\] ◻
By Proposition 11, in the threefold case we have: \[\label{first95collection95resolved95GM3} D^b(\widetilde{X}_3) = \langle j_*{\mathcal{O}}(-h), \widetilde{{\mathcal{K}}u}(X_3), {\mathcal{O}}, {\mathcal{U}}^\vee \rangle,\tag{9}\] while in the fourfold case we find:
\[\label{first95collection95resolved95GM4} D^b(\widetilde{X}_4) = \langle j_*{\mathcal{O}}(-2h), j_*{\mathcal{O}}(-h), \widetilde{{\mathcal{K}}u}(X_4), {\mathcal{O}},
{\mathcal{U}}^\vee, {\mathcal{O}}(H), {\mathcal{U}}^\vee(H) \rangle,\tag{10}\] and in dimension five: \[\label{first95collection95resolved95GM5}
D^b(\widetilde{X}_5) = \langle j_*{\mathcal{O}}(-3h), j_*{\mathcal{O}}(-2h), j_*{\mathcal{O}}(-h), \widetilde{{\mathcal{K}}u}(X_5), {\mathcal{O}}, {\mathcal{U}}^\vee, {\mathcal{O}}(H), {\mathcal{U}}^\vee(H), {\mathcal{O}}(2H), {\mathcal{U}}^\vee(2H)
\rangle.\tag{11}\]
Lemma 12. Let \({\mathcal{F}}\) be a vector bundle. Then one has: \[{\mathbb{R}}_{j_*{\mathcal{O}}}\phi^*{\mathcal{F}}\simeq \phi^*{\mathcal{F}}(-E).\]
Proof. Call \(r\) the rank of \({\mathcal{F}}\). Then, we have: \[\begin{align} \operatorname{Hom}_{\widetilde{X}}^\bullet(\phi^*{\mathcal{F}},
j_*{\mathcal{O}}) &\simeq H^\bullet(X, {\mathcal{F}}^\vee\otimes\phi_*j_*{\mathcal{O}}) \\ & \simeq H^\bullet(p, {\mathcal{F}}^\vee|_p) \\ & \simeq {\mathbb{C}}^r. \end{align}\] Therefore, the mutation, up to shift, is isomorphic to the
cone of a morphism \(\phi^*{\mathcal{F}}\longrightarrow j_*{\mathcal{O}}^{\oplus r}\). However, one has a short exact sequence: \[\label{eq:koszul95exc95divisor95times-F} 0\longrightarrow\phi^*{\mathcal{F}}(-E)\longrightarrow\phi^*{\mathcal{F}}\longrightarrow\psi^*{\mathcal{F}}\otimes j_*{\mathcal{O}}\longrightarrow 0\tag{12}\] and this concludes
the proof once we observe that \(\psi^*{\mathcal{F}}\otimes j_*{\mathcal{O}}\simeq j_*{\mathcal{O}}^{\oplus r}\). ◻
Lemma 13. One has \({\mathbb{L}}_{\mathcal{O}}j_*{\mathcal{O}}\simeq {\mathcal{O}}(-E)\), up to a shift.
Proof. One has \({\mathbb{L}}_{\mathcal{O}}j_*{\mathcal{O}}\simeq\operatorname{Cone}({\mathcal{O}}\longrightarrow j_*{\mathcal{O}})\). Then, the proof follows by the short exact sequence obtained by substituting
\({\mathcal{F}}\simeq {\mathcal{O}}\) in Equation 12 . ◻
Lemma 14. One has \({\mathbb{L}}_{\mathcal{O}}{\mathcal{U}}^\vee(-E) \simeq {\mathcal{O}}(-H+2E)\).
Proof. First, \(\operatorname{Hom}_{\widetilde{X}}({\mathcal{O}}, {\mathcal{U}}^\vee(-E)) \simeq {\mathbb{C}}^3[0]\) due to the sequence \[0\longrightarrow{\mathcal{U}}^\vee(-E)\longrightarrow{\mathcal{U}}^\vee\longrightarrow{\mathcal{U}}^\vee \otimes j_*{\mathcal{O}}\longrightarrow 0,\] again by Equation 12 .
Indeed, since \(\pi:\widetilde{X_n}\rightarrow X_n\) is the resolution of singularities, which is a birational morphism, we have \(\mathrm{Hom}_{\widetilde{X}_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}(-E))\cong\mathrm{Hom}_{X_n} (\mathop{\mathrm{\mathcal{O}}},\mathcal{U}\otimes I_p)\). Consider the standard short exact sequence \[0\rightarrow I_p\rightarrow\mathop{\mathrm{\mathcal{O}}}_{X_n}\rightarrow\mathop{\mathrm{\mathcal{O}}}_p\rightarrow 0.\] By [45], tensoring with \(\widetilde{{\mathcal{U}}}^{\vee}\), we get the following sequence on \(X\): \[0\rightarrow\mathcal{U}^{\vee}\otimes I_p\rightarrow\mathcal{U}^{\vee}\rightarrow\mathcal{U}^{\vee}\otimes\mathop{\mathrm{\mathcal{O}}}_p\rightarrow 0.\] Apply \(\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},-)\) to the short exact sequence, we have a long exact sequence \[0\rightarrow\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}\otimes
I_p)\rightarrow\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee})\rightarrow\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}|_p)\rightarrow\mathrm{Ext}^1_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}\otimes
I_p)\rightarrow 0,\] since \(\operatorname{Ext}^{>0}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee})=0\) by [45]. Note that \(\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee})=\mathbb{C}^5\) and \(\mathcal{U}^{\vee}|_p\cong\mathbb{C}^2\), then
\(\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}\otimes I_p)\geq 3\). We claim that \(\mathrm{Hom}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}\otimes I_p)\leq
3\), otherwise, the point \(p\) would be contained in a zero locus of four independent section of the vector bundle \(\mathcal{U}^{\vee}\), which is an empty set. Thus \(\mathrm{Ext}^{\geq 1}_{X_n}(\mathop{\mathrm{\mathcal{O}}},\mathcal{U}^{\vee}\otimes I_p)=0\), and the desired result is obtained.
Hence, we have: \[{\mathbb{L}}_{\mathcal{O}}{\mathcal{U}}^\vee(-E) = \operatorname{Cone}\left({\mathcal{O}}^{\oplus 3}\longrightarrow{\mathcal{U}}^\vee(-E)\right).\] By applying the snake lemma to the following diagram
we can deduce that the map is surjective, and therefore the kernel is a line bundle. \[\begin{tikzcd} {\mathcal{O}}(-E)^{\oplus 2}\ar[hook]{r} & {\mathcal{O}}^{\oplus 2} \ar[two heads]{r} & {\mathcal{U}}^\vee|_E \\
{\mathcal{Q}}^\vee \ar[hook]{r} & {\mathcal{O}}^{\oplus 5} \ar[two heads]{u} \ar[two heads]{r} & {\mathcal{U}}^\vee \ar[two heads]{u} \\ & {\mathcal{O}}^{\oplus 3} \ar[hook]{u} \ar{r} & {\mathcal{U}}^\vee(-E) \ar[hook]{u}
\end{tikzcd}\] The claim, hence, follows by the fact that \(c_1 ({\mathcal{U}}(E)) = -H + 2E\). ◻
Proposition 15. Let \(X_3\) be a nodal GM threefold and \(\widetilde{X}_3\) the resolution of the blowup in the node. Then one has: \[D^b(\widetilde{X}_3) = \langle {\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_3), {\mathcal{O}}(-2h+H), {\mathcal{O}}, {\mathcal{O}}(2h-H) \rangle\] Let now \(X_4\) be a nodal GM
fourfold. Then: \[D^b(\widetilde{X}_4) = \langle {\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_4), {\mathcal{O}}(-2h+H), {\mathcal{O}}, {\mathcal{O}}(2h-H), {\mathcal{O}}(-h+H), {\mathcal{O}}(h),
{\mathcal{O}}(3h-H)\rangle\] Finally, if \(X_5\) is a nodal GM fivefold, we obtain: \[\begin{align} D^b(\widetilde{X}_5) = & \langle
{\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_5), {\mathcal{O}}(-2h+H), {\mathcal{O}}, {\mathcal{O}}(2h-H), \\ &{\mathcal{O}}(-h+H), {\mathcal{O}}(h), {\mathcal{O}}(3h-H), {\mathcal{O}}(H), {\mathcal{O}}(2h), {\mathcal{O}}(4h-H)\rangle.
\end{align}\]
Proof. Let us start with the threefold case. In the SOD 9 we apply the inverse Serre functor to \(j_*{\mathcal{O}}(-h)\), obtaining: \[D^b(\widetilde{X}_3) = \langle \widetilde{{\mathcal{K}}u}(X_3), {\mathcal{O}}, {\mathcal{U}}^\vee, j_*{\mathcal{O}}\rangle.\] Then, by Lemma 12 and Lemma 13, we get: \[D^b(\widetilde{X}_3) = \langle \widetilde{{\mathcal{K}}u}(X_3), {\mathcal{O}}(-E), {\mathcal{O}}, {\mathcal{U}}^\vee(-E)\rangle.\] Then, we move \(\widetilde{\mathcal{K}}u(X_3)\) one step to
the right and we apply the inverse Serre functor to \({\mathcal{O}}(-E)\) finding: \[D^b(\widetilde{X}_3) = \langle {\mathbb{R}}_{ {\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_3), {\mathcal{O}},
{\mathcal{U}}^\vee(-E), {\mathcal{O}}(-E+h)\rangle.\] Let us move \({\mathcal{U}}^\vee(-E)\) one step to the left. By Lemma 14 and \(E = H-h\) we find: \[D^b(\widetilde{X}_3) = \langle {\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_3),
{\mathcal{O}}(h-2H), {\mathcal{O}}, {\mathcal{O}}(-2h+H)\rangle.\] Let us now consider the fourfold case, which can be addressed similarly. Again we start by mutating in Equation 10
the torsion objects to the end of the SOD finding: \[D^b(\widetilde{X}_4) = \langle \widetilde{{\mathcal{K}}u}(X_4), {\mathcal{O}}, {\mathcal{U}}^\vee, {\mathcal{O}}(H), {\mathcal{U}}^\vee(H), j_*{\mathcal{O}},
j_*{\mathcal{O}}(h) \rangle.\] Now, if we move \(j_*{\mathcal{O}}\) four steps to the right by Lemma 12 and then we mutate it through \({\mathcal{O}}\) via Lemma 13 we find \[D^b(\widetilde{X}_4) = \langle \widetilde{{\mathcal{K}}u}(X_4), {\mathcal{O}}(-E), {\mathcal{O}}, {\mathcal{U}}^\vee(-E), {\mathcal{O}}(H-E),
{\mathcal{U}}^\vee(H-E), j_*{\mathcal{O}}(h) \rangle.\] Since \(H-E = h\), we can mutate \(j_*{\mathcal{O}}(h)\) one step to the left with Lemma 12 and then an additional step to the left with Lemma 13: \[D^b(\widetilde{X}_4) = \langle \widetilde{{\mathcal{K}}u}(X_4),
{\mathcal{O}}(-E), {\mathcal{O}}, {\mathcal{U}}^\vee(-E), {\mathcal{O}}(h-E), {\mathcal{O}}(h), {\mathcal{U}}^\vee(h-E) \rangle.\] Now let us mutate the first block one step to the right, and \({\mathcal{O}}(-E)\) to
the end via inverse Serre functor: \[D^b(\widetilde{X}_4) = \langle {\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_4), {\mathcal{O}}, {\mathcal{U}}^\vee(-E), {\mathcal{O}}(h-E), {\mathcal{O}}(h),
{\mathcal{U}}^\vee(h-E), {\mathcal{O}}(h-E) \rangle.\] We conclude by applying Lemma 14 to both \({\mathcal{U}}^\vee(-E)\) and \({\mathcal{U}}^\vee(h-E)\), finding: \[D^b(\widetilde{X}_4) = \langle {\mathbb{R}}_{{\mathcal{O}}(-E)}\widetilde{{\mathcal{K}}u}(X_4),
{\mathcal{O}}(-H+2E),{\mathcal{O}},{\mathcal{O}}(h-E),{\mathcal{O}}(h-H+2E), {\mathcal{O}}(h), {\mathcal{O}}(h-E) \rangle.\] The claim follows by substituting \(E = H-h\) in the last SOD.
It is clear that the sequence of mutations for the fourfold case consist of iterating the threefold approach twice, up to twists by \({\mathcal{O}}(h)\). Unsurprisingly, the fivefold case can be addressed by applying the
same set of mutations three times. ◻
Let \(E\) be a globally generated vector bundle of rank \(m+2\) on a smooth projective variety \(B\), and \(\eta:{\mathbb{P}}(E^\vee)\longrightarrow B\) the associated projective bundle. Fix a relative hyperplane bundle \({\mathcal{L}}\) by the condition \(\eta_*{\mathcal{L}}\simeq E\). Given a general section \(\sigma\in H^0 ( {\mathbb{P}}(E^\vee), {\mathcal{L}}^{\otimes 2})\), call \(R = {\mathbb{V}}(\sigma)\)
its zero locus, and \(\psi: R\longrightarrow B\) the associated quadric fibration of relative dimenson \(m\). We call \({\mathcal{C}}_m\) the even part of
the Clifford algebra of \(R\), in the sense of [19]. Then, one has the following semiorthogonal decomposition [19]: \[\label{eq:sod95quadroc95fibration95in95general} D^b(R) = \langle D^b(B, {\mathcal{C}}_m),
\psi^*D^b(B)\otimes{\mathcal{L}}, \dots, \psi^*D^b(B)\otimes{\mathcal{L}}^{\otimes(m)}\rangle\tag{13}\]
where \(D^b(B, {\mathcal{C}}_m)\) denotes the derived category of coherent sheaves of modules over \({\mathcal{C}}_m\). In the following, for the quadric fibration \(R_n\), let us fix the notation \({\mathcal{B}}_n:= D^b({\mathbb{P}}^2, {\mathcal{C}}_n)\otimes{\mathcal{O}}(-h)\) where \({\mathcal{C}}_n\) is the even part of
the appropriate Clifford algebra. Then, \(R_n\) comes with a semiorthogonal decomposition as follows :
\[\label{eq:SOD95quadric95fibration} \begin{align} D^b(R_3) &= \langle {\mathcal{B}}_3, \psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W}))\rangle \text{(threefold)} \\
D^b(R_4) &= \langle {\mathcal{B}}_4, \psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W})), \psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W}))\otimes{\mathcal{O}}(h)\rangle \text{(fourfold)}\\ D^b(R_5) &= \langle {\mathcal{B}}_5,
\psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W})), \psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W}))\otimes{\mathcal{O}}(h), \psi^*D^b({\mathbb{P}}(\wedge^2\EuScript{W}))\otimes{\mathcal{O}}(2h)\rangle \text{(fivefold)} \end{align}\tag{14}\] where in
each of the decompositions, the component \({\mathcal{B}}_n\) is the derived category of modules over the even part of the Clifford algebra associated to \(R_n\).
Theorem 4. (Theorem 1) Let \(X_n(3\leq n\leq 5)\) be an ordinary Gushel-Mukai \(n\)-fold with a single node \(p\in X_n\). Then the categorical resolution\(\widetilde{{\mathcal{K}}u}(X_n)\) is given by \[\widetilde{{\mathcal{K}}u}(X_n)\simeq D^b(\mathbb{P}^2,\mathcal{C}_n),\] where \(D^b(\mathbb{P}^2,\mathcal{C}_n)\) is the derived category of coherent sheaves of modules on the even part of the
Clifford algebra associated to a quadric fibration \(R_n\) over \(\mathbb{P}^2\) of relative dimension \(n-2\). In particular, \(\widetilde{{\mathcal{K}}u}(X_4)\simeq D^b(\mathbb{P}^2,\mathcal{C}_4)\simeq D^b(S,\alpha)\), where \(S\) is a degree two \(K3\) surface and \(\alpha\in\mathrm{Br}(S)\) is the Brauer class.
Proof. Since \(\widetilde{X}_n\) and \(R_n\) are related by an Atiyah flop (Diagram 5 ), the functor \(q_*p^*\)
is an equivalence of categories [46]. Note that by the projection formula one has \(q_*p^*{\mathcal{O}}(a(H-2E) + bh) =
{\mathcal{O}}(a(H-2E) + bh)\) for all integers \(a, b\), since \(H-2E = -H+2h\) is the pullback of the hyperplane class from \({\mathbb{P}}(\wedge^2\EuScript{W})\) (see Lemma 4).
Now, by Beilinson’s semiorthogonal decomposition for the projective space [47], one has \(D^b({\mathbb{P}}(\wedge^2\EuScript{W})) = \langle{\mathcal{O}}(-2h+H), {\mathcal{O}}, {\mathcal{O}}(2h-H)\rangle\). Then, the desired equivalence follows by comparing Equation 14
with the image under \(q_*p^*\) of the semiorthogonal decompositions of Proposition 15. Finally, the
equivalence \(D^b(\mathbb{P}^2,\mathcal{C}_4)\simeq D^b(S,\alpha)\) is a consequence of [48]. ◻
Remark 16. Let \(n\) be odd. Then, the semiorthogonal decomposition of Equation ?? can be further refined to the following: \[\label{eq:sod95resolved95gm95nfold95alternative} \begin{align} D^b(\widetilde{X}_n) = \langle & {\mathcal{O}}_E((n-2)E), \dots, {\mathcal{O}}_E(2E), {\mathcal{S}}_E(E), {\mathcal{O}}_E(E), \\ &
\widetilde{{\mathcal{K}}u}(X_n), {\mathcal{O}}, {\mathcal{U}}^\vee, \dots, {\mathcal{O}}((n-3)H), {\mathcal{U}}^\vee((n-3)H)\rangle, \end{align}\tag{15}\] where \({\mathcal{S}}_E\) is the spinor bundle of the
exceptional quadric \(E\). This follows from applying [17] to
the following semiorthogonal decomposition: \[D^b(E) = \langle {\mathcal{O}}_E((n-2)E), \dots, {\mathcal{O}}_E(2E), {\mathcal{S}}_E(E), {\mathcal{O}}_E(E), {\mathcal{S}}_E, {\mathcal{O}}_E \rangle.\] Then, by replacing
Equation ?? with 15 in the proof of Proposition 15 for \(n\) odd, we obtain: \[D^b(\mathbb{P}^2,\mathcal{C}_n) \simeq \langle {\mathcal{F}}, \widetilde{{\mathcal{K}}u}(X_n) \rangle,\] where \({\mathcal{F}}\) is an
exceptional object, This phenomenon will be object of further study.
Kuznetsov and Perry have conjectured the existence of a relation between Kuznetsov components of Gushel–Mukai varieties of different dimensions [11]. Such conjecture has been proven later by the same authors, in the smooth case, with the tool of categorical joins. In particular, one has the following:
Theorem 5. [49]Let \(Y_n\) be a smooth Gushel–Mukai variety
of dimension \(n\), with \(5\leq n\leq 6\). Then there is a smooth Gushel–Mukai \(n-2\)-fold \(Y_{n-2}\) such that \({\mathcal{K}}u(Y_n) \simeq {\mathcal{K}}u(Y_{n-2})\).
In the following, as an application of Theorem 4, we extend Theorem 5 to the case of one-nodal ordinary Gushel–Mukai varieties, with a codimension two condition on the choice of the nodal quadric.
Hyperbolic reduction is a classical construction, let us summarize it here. Let \(Q\subset{\mathbb{P}}^n\) be a quadric hypersurface and \(x\in Q\). Consider the affine tangent bundle
\(\widetilde{T}\) described by the following short exact sequence: \[0\longrightarrow{\mathcal{O}}(-1)\longrightarrow\widetilde{T}\longrightarrow T(-1)\longrightarrow 0\] where \(T\) is the tangent bundle. Call \({\mathbb{P}}(\widetilde{T}_x)\simeq{\mathbb{P}}^{n-1}\) the embedded tangent space (the projectivization of the fiber over \(x\) of the affine tangent bundle). Then, the projection \(\pi_x:{\mathbb{P}}^{n}\dashrightarrow {\mathbb{P}}^{n-1}\) maps \(Q\setminus x\) to a quadric \(Q_x\), isomorphic to \(Q\cap{\mathbb{P}}(\widetilde{T}_x)\). The hyperbolic reduction \(\widetilde{Q}\) of \(Q\) with respect to
\(x\) is a quadric of dimension \(n-3\) obtained as the projection of \(Q_x\) from its vertex. If \(Q\) is smooth, \(\widetilde{Q}\) is smooth as well.
This construction has been described in families as follows (see [26] for more
details, and a more general construction): let \({\mathcal{E}}\) be a globally generated vector bundle of rank \(n\) over a smooth projective base \(S\),
call \(p:{\mathbb{P}}({\mathcal{E}}^\vee)\longrightarrow S\) the associated projective bundle map. Let \({\mathcal{L}}\) be the line bundle restricting to \({\mathcal{O}}(1)\) to each fiber, and such that \(p_*{\mathcal{L}}\simeq {\mathcal{E}}\). A quadric fibration \(R\) over \(S\),
of relative dimension \(n-2\), will be the zero locus of a section \(\sigma\in H^0({\mathbb{P}}({\mathcal{E}}^\vee), {\mathcal{L}}^{\otimes 2})\). Clearly, one can visualize \(\sigma\) as a symmetric morphism \({\mathcal{E}}^\vee\longrightarrow{\mathcal{E}}\), which we will still denote by \(\sigma\).
Now, we say that a subbundle \({\mathcal{F}}\subset {\mathcal{E}}\) is \(\sigma\)-isotropic if \(\sigma|_{\mathcal{F}}= 0\). Let us take a \(\sigma\)-isotropic line bundle \({\mathcal{F}}\subset{\mathcal{E}}\). Then, the hyperbolic reduction associated to \(R\) and \({\mathcal{F}}\) is given by the vanishing of the restriction of \(\sigma\) to \({\mathcal{F}}^\perp / {\mathcal{F}}\), i.e. a quadric fibration of relative
dimension \(n-4\). It is a known fact that the operation of hyperbolic reduction in families preserves the discriminant divisor in \(S\)[26].
Let us specialize this picture to our setting. Consider the vector bundle \({\mathcal{E}}:= \EuScript{P}^\vee\otimes T(-\zeta)\oplus{\mathcal{O}}(\zeta)\) on \({\mathbb{P}}(\wedge^2\EuScript{W})\). This is a globally generated, homogeneous vector bundle of rank five. Then, as a divisor in \({\mathbb{P}}({\mathcal{E}}^\vee)\), one has \(R_5 = 2 h\): that is, one has \(R_5 \simeq{\mathbb{V}}(\sigma)\subset{\mathbb{P}}({\mathcal{E}}^\vee)\) where \(\sigma\in H^0({\mathbb{P}}({\mathcal{E}}^\vee),
{\mathcal{O}}(2 h ))\simeq H^0({\mathbb{P}}(\wedge^2\EuScript{W}), \mathop{\mathrm{Sym}}^2{\mathcal{E}})\).
On the other hand, for \(2\leq n \leq 4\), one has that \(R_n\) is the zero locus of an element of \(H^0({\mathbb{P}}(\widetilde{\mathcal{E}}_n^\vee,
{\mathcal{O}}(2 h ))\simeq H^0({\mathbb{P}}(\wedge^2\EuScript{W}), \mathop{\mathrm{Sym}}^2\widetilde{\mathcal{E}}_k)\) where \(\widetilde{\mathcal{E}}_n\) is a rank \(n\) vector
bundle defined by the following exact sequences (see the proof of Lemma 5):
Proposition 17. Fix an embedding \(\iota: {\mathcal{O}}^{\oplus 2}\xhookrightarrow{\,\,\,\,\,}{\mathcal{E}}\). Then, there is a codimension two subspace \({\mathcal{H}}_\iota\subset H^0({\mathbb{P}}(\wedge^2 \EuScript{W}),\mathop{\mathrm{Sym}}^2{\mathcal{E}})\) such that for every \(\sigma\in\EuScript{H}_\iota\), the embedding \(\iota\) is \(\sigma\)-isotropic.
Proof. Pick a \(\sigma\) and consider an embedding: \[\iota : {\mathcal{O}}^{\oplus 2}\xhookrightarrow{\,\,\,\,\,}{\mathcal{E}}.\] This induces a local decomposition of \(\sigma\) (seen as a symmetric morphism \({\mathcal{E}}^\vee\longrightarrow{\mathcal{E}}\)) as follows: \[\sigma \simeq \left( \begin{array}{ccc} \sigma_{11} &
\sigma_{12} & \sigma_{13} \\ \sigma_{12} & \sigma_{22} & \sigma_{23} \\ \sigma_{13} & \sigma_{23} & \sigma_{33} \end{array} \right)\] where \(\sigma_{11}, \sigma_{22}, \sigma_{12}\) are scalars,
\(\sigma_{13}\) and \(\sigma_{23}\) are morphisms \({\mathcal{O}}\longrightarrow\widetilde{\mathcal{E}}_3\), and \(\sigma_{33}\) is a symmetric morphism \(\widetilde{\mathcal{E}}_3^\vee\longrightarrow\widetilde{\mathcal{E}}_3\). For a fixed \(\iota\), the \(\sigma\)-isotropy condition on \(\iota\) is a codimension two condition (\(\sigma_{11} = \sigma_{22}=0\)) in the space \(H^0({\mathbb{P}}(\wedge^2 \EuScript{W}),\mathop{\mathrm{Sym}}^2{\mathcal{E}})\). ◻
Identically, one proves the following:
Proposition 18. Fix an embedding \(\iota: {\mathcal{O}}^{\oplus 2}\xhookrightarrow{\,\,\,\,\,}\widetilde{\mathcal{E}}_4\). Then, there is a codimension two subspace \(\widetilde{\mathcal{H}}_\iota\subset H^0({\mathbb{P}}(\wedge^2 \EuScript{W}),\mathop{\mathrm{Sym}}^2\widetilde{\mathcal{E}}_4)\) such that for every \(\sigma\in\widetilde{\EuScript{H}}_\iota\),
the embedding \(\iota\) is \(\sigma\)-isotropic.
Now, observe that embeddings \({\mathcal{O}}\xhookrightarrow{\,\,\,\,\,}{\mathcal{E}}\) are in one-to-one correspondence with hyperplane sections of \(\operatorname{Gr}\) passing through
the node. In fact, such a hyperplane section is the premiage of a general hyperplane section of \({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus\wedge^2\EuScript{P})\) with respect to the projection from \({\mathbb{P}}(\wedge^2\EuScript{V})\), and one has \(H^0({\mathbb{P}}(\EuScript{P}\otimes\EuScript{W}\oplus\wedge^2\EuScript{P}), {\mathcal{O}}(h))\simeq H^0({\mathbb{P}}(\wedge^2\EuScript{W}),
{\mathcal{E}}^\vee)\) (recall that \({\mathcal{O}}(h)\) is a relative hyperplane bundle of \({\mathbb{P}}({\mathcal{E}}^\vee)\)). Therefore, the choice of a nodal Gushel–Mukai variety
of dimension \(n<5\) defines a morphism from a trivial bundle of rank \(5-n\) to \({\mathcal{E}}\) as in Equation 16 .
Theorem 6. (Theorem 2 (1)). Choose two general hyperplane sections \(s_1, s_2\) of \(\operatorname{Gr}\) passing through the node. Then, in the space of quadric hypersurfaces in \({\mathbb{P}}(\wedge^2\EuScript{V})\) passing through the node, there is a codimension two subspace
\(\EuScript{H}\) such that for every \(\EuScript{Q}\in\EuScript{H}\), the Gushel–Mukai fivefold \(X_5:= \operatorname{Gr}\cap\EuScript{Q}\) enjoys the
following property: there is a nodal Gushel–Mukai threefold \(X_3 = {\mathbb{V}}(s_1, s_2)\cap X_5\) such that \(\widetilde{{\mathcal{K}}u}(X_3)\simeq \widetilde{{\mathcal{K}}u}(X_5)\).
Proof. The image of a quadric \(\EuScript{Q}\) containing the node, through the projection from the node, is given by the vanishing of a general \(\sigma\in |2h|\). Similarly,
the images of \(s_1, s_2\) will be general elements of \(|h|\): call them \(\bar s_1, \bar s_2\). Denote by \(\iota\) the
embedding \({\mathcal{O}}^{\oplus 2}\longrightarrow{\mathcal{E}}\) defined by \(\bar s_1, \bar s_2\). Then, by Proposition 17, there is a codimension two space of quadrics \(\EuScript{H}:=\EuScript{H}_\iota\) such that for every \(\sigma\in
\EuScript{H}_\iota\), the embedding \(\iota\) is \(\sigma\)-isotropic. Let us now fix \(R_5:=
{\mathbb{V}}(\sigma)\subset{\mathbb{P}}({\mathcal{E}}^\vee)\) and \(R_3:= {\mathbb{V}}(\sigma, \bar s_1, \bar s_2)\in{\mathbb{P}}({\mathcal{E}}^\vee)\) (or equivalently, \(R_3 =
{\mathbb{V}}(\sigma)\subset{\mathbb{P}}({\mathcal{E}}_3^\vee)\), where the \({\mathcal{E}}_3\) is the cokernel of \(\iota\)). Then, since \(\iota\) is
\(\sigma\)-isotropic, \(R_3\) is by definition a hyperbolic reduction of \(R_5\). Let us now introduce \(X_5:=\operatorname{Gr}\cap\EuScript{Q}\) and \(X_3 := X_5\cap{\mathbb{V}}(s_1, s_2)\). Then, by Theorem 4, one has \(\widetilde{K} u(X_5) \simeq D^b({\mathbb{P}}^2, \mathcal{C}_5)\) and \(\widetilde{{\mathcal{K}}u}(X_3) \simeq
D^b({\mathbb{P}}^2, \mathcal{C}_3)\), where the right-hand sides are the derived categories of modules over the even part of the Clifford algebras associated, respectively, to \(R_5\) and \(R_3\). But since \(R_3\) is a hyperbolic reduction of \(R_5\), one has \(D^b({\mathbb{P}}^2, \mathcal{C}_5)\simeq D^b({\mathbb{P}}^2,
\mathcal{C}_3)\), and this concludes the proof. ◻
A similar result, relating GM varieties of dimension four and two, can be proven in exactly the same way, due to Proposition 18:
Theorem 7. (Theorem 2 (2)). Choose three general hyperplane sections \(s_1, s_2, s_3\) of \(\operatorname{Gr}\) passing through the node \(p\). Then, in the space of quadric hypersurfaces in \({\mathbb{P}}(\wedge^2\EuScript{V})\) which are nodal in
\(p\), there is a codimension two subspace \(\EuScript{H}\) such that for the general \(\EuScript{Q}\in\EuScript{H}\), the Gushel–Mukai fourfold \(X_4:= \operatorname{Gr}\cap\EuScript{Q}\cap{\mathbb{V}}(s_1)\) enjoys the following property: there is a smooth \(K3\) surface \(\widetilde{X}_2\) of degree two,
which is the blowup of \(X_2 = {\mathbb{V}}(s_2, s_3)\cap X_4\) in \(p\), such that \(D^b(\widetilde{X}_2)\simeq \widetilde{{\mathcal{K}}u}(X_4)\).
While rationality is known in the case of fivefolds, it is not obvious for fourfolds. However, choosing the quadric \(\EuScript{Q}\) as in 7 forces \(X_4\) to be rational:
Corollary 2. Let \(X_4\) be a nodal GM fourfold satisfying the assumptions of Theorem 7. Then, \(X_4\) is rational.
Proof. Since \(R_4\) admits a hyperbolic reduction, a birational map to \({\mathbb{P}}^4\) can be constructed by projecting from an isotropic section. Therefore, \(X_4\) and \(\widetilde{X}_4\) are also rational. ◻
4.2 The relationship with Verra threefolds and fourfolds↩︎
A Verra threefold \(V_3\) is a \((2,2)\)-section of \({\mathbb{P}}^2\times{\mathbb{P}}^2\) (see [23]). A general smooth Verra threefold admits two conic fibrations over \({\mathbb{P}}^2\) with discriminant divisor given by smooth sextic curves, and therefore it
admits a semiorthogonal decomposition as follows:
where \({\mathcal{K}}u(V_3)\) is the “Clifford component” and \({\mathcal{O}}(1)\) is the hyperplane bundle of \({\mathbb{P}}^2\). Similarly, Verra
fourfolds are double covers of \({\mathbb{P}}^2\times{\mathbb{P}}^2\) branched over a Verra threefold and they also admit two quadric fibrations over \({\mathbb{P}}^2\) branched along a
sextic curve [24]. Hence, for a Verra fourfold \(V_4\), we can define \({\mathcal{K}}u(V_4)\) by its structure as a quadric fibration as for threefolds.
In this section, we illustrate how GM threefolds satisfying the assumptions of Theorem 6 are related to Verra threefolds by the fact that the corresponding
fibration \(R^3\) is a hyperbolic reduction of a quadric fibration \(W\longrightarrow{\mathbb{R}}^2\), which also admits a hyperbolic reduction to a Verra threefold. The same result for
fourfolds has been obtained by [22]. A natural consequence of these results is an equivalence of the corresponding Kuznetsov components.
Proposition 19. Take a general, nodal GM threefold \(X_3\), and call \(R_3\) the image of \(X_3\) through the flop of Diagram 5 . Then the following holds:
There is a quadric fibration \(W\longrightarrow{\mathbb{P}}^2\), and a Verra threefold \(V_3\), such that both \(R_3\) and \(V_3\) are hyperbolic reductions of \(W\).
Proof. Let us recall that \(R_3\) is the vanishing locus of a section of \({\mathcal{O}}(2h)\) on the projectivization of \(\widetilde{\mathcal{E}}_3^\vee\), where \(\widetilde{\mathcal{E}}_3 = T_{{\mathbb{P}}^2}(-\zeta)^{\oplus 2}\oplus{\mathcal{O}}(\zeta)/{\mathcal{O}}^{\oplus 2}\) (see Section 4).
In light of [22], let us construct a short exact sequence which resolves \(\widetilde{\mathcal{E}}_3\) by
split bundles. One has the following diagram: \[\begin{tikzcd} 0\ar{d}\ar{r} & {\mathcal{O}}(-\zeta)^{\oplus 2}\ar[hook]{d} & \\ {\mathcal{O}}^{\oplus 2}\ar[hook]{r}\ar[equals]{d} & {\mathcal{O}}^{\oplus
6}\oplus{\mathcal{O}}(\zeta) \ar[two heads]{r}\ar[two heads]{d} & {\mathcal{O}}^{\oplus 4}\oplus{\mathcal{O}}(\zeta) \\ {\mathcal{O}}^{\oplus 2}\ar[hook]{r} & T_{{\mathbb{P}}^2}(-\zeta)^{\oplus 2}\oplus{\mathcal{O}}(\zeta) \ar[two heads]{r} &
\widetilde{\mathcal{E}}_3 \end{tikzcd}\] where the middle vertical sequence follows from the Euler sequence of \({\mathbb{P}}^2\). Then, by the snake lemma applied to the two leftmost vertical sequences, we get the
desired split resolution: \[0\longrightarrow{\mathcal{O}}(-\zeta)^{\oplus 2}\longrightarrow{\mathcal{O}}^{\oplus 4}\oplus{\mathcal{O}}(\zeta) \longrightarrow\widetilde{\mathcal{E}}_3\longrightarrow 0.\] Since \(H^2({\mathbb{P}}^2, {\mathcal{O}}(-2\zeta)) = 0\), we can apply [22] to deduce the following: in the projective
bundle \({\mathbb{P}}({\mathcal{O}}^{\oplus 4}\oplus{\mathcal{O}}(-\zeta)^{\oplus 3})\), there is a quadric fibration \(W\) of relative dimension five, such that \(R_3\) is a hyperbolic reduction of \(W\). Then, by [22], \(W\) admits a hyperbolic reduction to a conic bundle in \({\mathbb{P}}({\mathcal{O}}(-\zeta)^{\oplus 3})\), which is exactly the Verra threefold \(V_3\). ◻
Corollary 3. (Proposition 1) Let \(X_n\) be a one-nodal Gushel–Mukai variety of dimension three or four. Then
there is a Verra variety \(V_n\) of the same dimension such that \(\widetilde{{\mathcal{K}}u}(X_n)\simeq {\mathcal{K}}u(V_n)\).
Proof. This is a direct consequence of Proposition 19 and [26] for threefolds, and [22] and [26] for fourfolds. ◻
Remark 20. As the geometric resolution of a rational nodal Gushel-Mukai fourfold considered in Theorem 7 is birationally equivalent to a Verra
fourfold \(V\), thus \(V\) is rational. As a result, Theorem 7 and Corollary 3 imply that the Kuznetsov component \({{\mathcal{K}}u}(V)\simeq D^b(\widetilde{X_2})\). This in turn, confirms an
instance of the Verra analogue of Kuznetsov’s rationality conjecture, which predicts that a Verra fourfold is rational if and only if the Kuznetsov component is derived equivalent to an actual \(K3\) surface.
5 Categorical Torelli theorem for nodal Gushel-Mukai threefolds↩︎
Torelli-type problems are a developing area that has attracted a lot of attention from the algebraic geometry community, and broadly consist in understanding the roles of Hodge strucrtures and derived categories as invariants on several classes of
varieties. While many counterexamples for several formulations of this problem have been found for Calabi–Yau manifolds [50]–[54], Fano varieties behave very rigidly from the point of view of derived categories. Hence, the usual formulation of the categorical Torelli theorem for
smooth Fano varieties addresses the problem of whether the Kuznetsov component of a Fano variety, instead of the whole derived category, characterizes its (birational) isomorphism class. In many cases, including smooth del Pezzo threefolds, Gushel-Mukai
varieties, Enriques surfaces, hypersurfaces in (weighted) projective spaces, the answer has been shown to be affirmative [28], [55]–[59]. Recently, the study of categorical Torelli problems has been extended to singular Fano
varieties. In particular, in [29], the authors show the categorical data of a smooth subcategory inside the Kuznetsov component of a
1-nodal non-factorial Fano threefolds characterizes its isomorphism class. In the present work we address nodal factorial Fano threefolds: here, instead of using the smooth subcategory, which is smaller than the Kuznetsov component, we use the
larger one, i.e. the Categorical resolution of singularities.
For a 1-nodal cubic threefold \(Y_3\), a categorical resolution \(\widetilde{{\mathcal{K}}u}(Y_3)\) contains a semi-orthogonal component as the derived category \(D^b(\Gamma_4)\) of a smooth curve \(\Gamma_4\) of genus four in \(\mathbb{P}^3\), and the complement is an exceptional object(See [60]). Note that the geometric resolution \(\widetilde{Y_3}\cong\mathrm{Bl}_{\Gamma_4}\mathbb{P}^3\). This means that \(\widetilde{{\mathcal{K}}u}(Y_3)\) characterizes the isomorphism class of the 1-nodal cubic threefold \(Y_3\) by [61]. In the case of 1-nodal Gushel-Mukai threefolds \(X_3\) considered in our article, as explained in Theorem 4, a categorical resolution \(\widetilde{{\mathcal{K}}u}(X_3)\) is given by
the category \(\mathcal{B}_3=D^b(\mathbb{P}^2,\mathcal{C}_3)\). As a corollary, we show \(\widetilde{{\mathcal{K}}u}(X_3)\) determines the birational class of \(X_3\). More precisely,
Theorem 8. (Theorem 3). Let \(X_3, X_3'\) be general 1-nodal Gushel-Mukai threefolds, assume
that there is an equivalence \(\Phi:\widetilde{{\mathcal{K}}u}(X_3)\simeq\widetilde{{\mathcal{K}}u}(X_3')\), then \(X_3\sim X_3'.\)
Lemma 21. Let \(R_3\) be the quadric fibration over \(\mathbb{P}^2\) of relative dimension one, i.e. a conic bundle. Then \(\mathrm{H}^1(R_3,\mathop{\mathrm{\mathcal{O}}}_{R_3})=0\).
Proof. Recall that \(R_3\) is the zero locus of a section of \({\mathcal{O}}(2h)\) on \({\mathbb{P}}(\widetilde{\mathcal{E}}_3^\vee)\).
Therefore, we have a short exact sequence on \({\mathbb{P}}(\widetilde{\mathcal{E}}_3^\vee)\): \[0\longrightarrow{\mathcal{O}}(-2h)
\longrightarrow{\mathcal{O}}\longrightarrow{\mathcal{O}}_{R_3}\longrightarrow 0\] from which we deduce that \(H^1( R_3, {\mathcal{O}}_{R_3}) \simeq H^2({\mathbb{P}}(\widetilde{\mathcal{E}}_3^\vee),
{\mathcal{O}}(-2h))\). We conclude once we see that \({\mathbb{P}}(\widetilde{\mathcal{E}}_3^\vee)\) is a \({\mathbb{P}}^2\)-bundle and \(h\) is a
relative hyperplane class, and therefore \({\mathcal{O}}(-2h)\) has no cohomology. ◻
Proof of Theorem 8. By Theorem 1 above,
there is a Fourier-Mukai equivalence \(\widetilde{{\mathcal{K}}u}(X_3)\simeq\)\(D^b(\mathbb{P}^2,\mathcal{C}_3)\), where the right-hand side is a Clifford component of a conic bundle \(R_3\) over \(\mathbb{P}^2\). Denote by \(C\) and \(C'\) the discriminant locus of the conic bundle \(R_3\) and \(R_3'\), respectively. Then by [62], such a component
is equivalent to the derived category \(D^b(\sqrt{\mathbb{P}^2, C},\alpha)\) of twisted sheaves on the root stack of \(\mathbb{P}^2\). Then \(\Phi\) becomes
an equivalence \(\Psi:D^b(\sqrt{\mathbb{P}^2,C'},\alpha)\simeq D^b(\sqrt{\mathbb{P}^2,C},\alpha').\) Combining the argument in [63] and [64], such an equivalence is given by a Fourier-Mukai functor1. Note that the semiorthogonal decomposition of \(R_3\) is given by \[D^b(R_3)=\langle
D^b(\mathbb{P}^2,\mathcal{C}_3),q^*D^b(\mathbb{P}^2)\rangle,\] where \(q:R_3\rightarrow\mathbb{P}^2\) is the quadric bundle map. As \(\mathrm{H}^1(R_3,\mathop{\mathrm{\mathcal{O}}}_{R_3})=0\) by Lemma 21. Then we apply
the abstract intermediate Jacobian construction defined in [65], we have a chain of isomorphisms: \[J(R_3)\cong\mathrm{J}(D^b(\mathbb{P}^2,\mathcal{C}_3))\cong\mathrm{J}(D^b(\mathbb{P}^2,\mathcal{C}_3'))\cong J(R_3')\] as p.p.a.v. Denote by \(\Gamma_6,\Gamma_6'\) the discriminant
locus of quadric bundle maps \(q:R_3\rightarrow\mathbb{P}^2\) and \(R_3'\rightarrow\mathbb{P}^2\), respectively. Then by [66], \(J(R_3)\cong\mathrm{Prym}(\widetilde{\Gamma_6},\Gamma_6)\) and \(J(R_3')\cong\mathrm{Prym}(\widetilde{\Gamma_6'},\Gamma_6')\). Then \(\mathrm{Prym}(\widetilde{\Gamma_6},\Gamma_6)\cong\mathrm{Prym}(\widetilde{\Gamma_6'},\Gamma_6')\) as
p.p.a.v. Note that \(\Gamma_6\) and \(\Gamma_6'\) are smooth sextic curve in \(\mathbb{P}^2\), the genus \(g(\Gamma_6)=g(\Gamma_6')=10\). Then by Prym-Torelli Theorem[67], we have \(\widetilde{\Gamma_6}/\Gamma_6\cong\widetilde{\Gamma_6'}/\Gamma_6'\) as double covers. This in turn implies that \(R_3\) is birationally equivalent to \(R_3'\) by [66]. By the birational equivalences \(R_3\sim\widetilde{X_3}\) and \(R_3'\sim\widetilde{X_3'}\), we get \(X_3\sim X_3'\). ◻
Remark 22. We sketch an alternate argument by observing that \(\widetilde{{\mathcal{K}}u}(X_3)\simeq{{\mathcal{K}}u}(V_3)\) for some Verra threefold as proved in Corollary 3. The equivalence \(\widetilde{{\mathcal{K}}u}(X_3)\simeq\widetilde{{\mathcal{K}}u}(X_3')\) implies that \({{\mathcal{K}}u}(V_3)\simeq{{\mathcal{K}}u}(V_3')\) and Verra threefolds \(V_3\) and \(V_3'\) are birationally equivalent to \(X_3\) and \(X_3'\), respectively. As a Verra threefold is Fano, apply the abstract intermediate Jacobian construction defined in [65], we get \(J(V_3)\cong J(V_3')\) as p.p.a.v. Then by Torelli theorem for Verra threefolds proved in [68], we get \(V\cong V'\) and then \(X_3\sim X_3'\).
6.1 Another birational model for the resolution of nodal Gushel–Mukai threefolds↩︎
In this section, using the technique established in [69], we will construct a conic bundle \(U\)
which is a spinor modification of \(R_3\). In particular, the Clifford component of its derived category will be equivalent to the one of \(R_3\), providing a new birational model of \(\widetilde{X_3}\).
Lemma 23. Consider a rational curve \(C\subset R_3\) which is contracted by the small contraction \(R_3\longrightarrow T_3\). Then there is a nontrivial extension: \[0\longrightarrow O(-h+2\zeta) \longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{I}}_{C|R_3}\longrightarrow 0.\] Moreover, \({\mathcal{F}}\) is an abstract spinor bundle in the sense of [69].
Proof. Call \(\eta\) the embedding of \(C\) in \(R_3\). One has: \[\begin{align}
R^\bullet\operatorname{Hom}(\eta_*{\mathcal{O}}_C, {\mathcal{O}}(-h+2\zeta)) & \simeq R^\bullet\operatorname{Hom}({\mathcal{O}}(-h+2\zeta), \eta_*{\mathcal{O}}_C[3])\\ & \simeq R^\bullet\operatorname{Hom}({\mathcal{O}}_C(2\zeta),
{\mathcal{O}}_C[3])\\ & \simeq H^\bullet({\mathbb{P}}^1, {\mathcal{O}}_{{\mathbb{P}}^1}(-2)[3]). \end{align}\] where in the second isomorphism we used the fact that \({\mathcal{O}}_C(-h)\simeq{\mathcal{O}}_C\)
(recall that \(C\) is contracted to a point by \(|h|\)). This proves the existence of a unique two-terms extension of \(\eta_*{\mathcal{O}}_C\) and \({\mathcal{O}}(-h+2\zeta)\). Together with the ideal sheaf sequence of \(C\) in \(R_3\), this proves the first claim.
To verify the second statement, we need to prove that:
\(\psi_*{\mathcal{F}}= 0\)
\(\det F = {\mathcal{O}}(-h+k\zeta)\) for some integer \(k\).
\((1)\) Applying the pushforward to the ideal sheaf sequence, we get: \[\psi_*{\mathcal{F}}= \operatorname{Cone}\left(
\psi_*{\mathcal{I}}_{C|R_3}[-1]\longrightarrow\psi_*{\mathcal{O}}(-h+2\zeta). \right)\] Let us treat the two terms separately. The first one is: \[f_*{\mathcal{I}}_{C|R_3} \simeq \operatorname{Cone}\left(
{\mathcal{O}}[-1]\longrightarrow\bar\eta_*{\mathcal{O}}_L[-1] \right) \simeq {\mathcal{O}}(-\zeta)\] where the term \(\bar\eta_*{\mathcal{O}}_L\) is due to the fact that \(L:=\psi(C)\) is a line, and the morphisms \(\psi\) and \(\eta\) give a commutative diagram as follows: \[\begin{tikzcd}[row sep =
huge, column sep = huge] C\ar[hook]{r}{\eta}\ar[swap]{d}{\bar\psi} & R_3 \ar{d}{\psi}\\ L\ar[hook]{r}{\bar\eta} & {\mathbb{P}}^2 \end{tikzcd}\] To address the second term, let us begin by noting that: \[{\mathcal{O}}(-h+2\zeta) \simeq {\mathcal{O}}(-\zeta)\otimes \psi^!{\mathcal{O}}[-1]\simeq {\mathcal{O}}(-\zeta)\otimes {\mathcal{H}}om ({\mathcal{O}}, \psi^!{\mathcal{O}}[-1]),\] where “!” denotes the right adjoint of the
derived pushforward. Then, by Grothendieck duality and projection formula, we see that \[\begin{align} \psi_* \left( {\mathcal{O}}(-\zeta)\otimes {\mathcal{H}}om ({\mathcal{O}}, \psi^!{\mathcal{O}}[-1]) \right) & \simeq
{\mathcal{O}}(-\zeta)\otimes \psi_* {\mathcal{H}}om ({\mathcal{O}}, \psi^!{\mathcal{O}}[-1]) \\ & \simeq {\mathcal{O}}(-\zeta)\otimes {\mathcal{H}}om ({\mathcal{O}}, {\mathcal{O}}[-1]) \\ & \simeq {\mathcal{O}}(-\zeta)[-1] \end{align}\]
Putting everything together, one gets \(\psi_*{\mathcal{F}}= 0\).
\((2)\) simply folloows by the short exact sequence: \[0\longrightarrow{\mathcal{O}}(-h+2\xi)\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{I}}_{C|R_3}\longrightarrow 0.\] ◻
Lemma 24. One has the following resolutions on \({\mathbb{P}}^2\): \[0\longrightarrow T(-\zeta)\oplus{\mathcal{O}}(\zeta)
\longrightarrow{\mathcal{E}}_3\longrightarrow\bar\eta_*{\mathcal{O}}_L\longrightarrow 0\]\[0\longrightarrow{\mathcal{O}}(-\zeta)\longrightarrow{\mathcal{O}}^{\oplus
3}\oplus{\mathcal{O}}(\zeta)\longrightarrow{\mathcal{E}}_3\longrightarrow\bar\eta_*{\mathcal{O}}_L\longrightarrow 0\]
Proof. The second resolution clearly comes from the first combined with the Euler sequence of \({\mathbb{P}}^2\), hence let us focus on the first one. By the definition of \({\mathcal{E}}_3\) and the Euler sequence, one can realize a commutative diagram: \[\begin{tikzcd} {\mathcal{O}}(-\zeta)\ar[hook]{d} \ar[hook]{r} & {\mathcal{O}}^{\oplus 3}\oplus
{\mathcal{O}}(\zeta) \ar[hook]{d} \ar[two heads]{r} & T(-\zeta)\oplus{\mathcal{O}}(\zeta) \\ {\mathcal{O}}^{\oplus 2}(-\zeta) \ar[two heads]{d} \ar[hook]{r} & {\mathcal{O}}^{\oplus 4}\oplus {\mathcal{O}}(\zeta) \ar[two heads]{d} \ar[two heads]{r}
& {\mathcal{E}}_3 \\ {\mathcal{O}}(-\zeta) & {\mathcal{O}}& \end{tikzcd}\] The desired sequence is obtained by snake lemma. ◻
Remark 25. Note that the second resolution is very similar to the ones obtained by Kuznetsov for other conic bundles over \({\mathbb{P}}^2\) (cf. [69]).
Proposition 26. Let us call \(U\) the conic bundle on \({\mathbb{P}}^2\) defined by the vanishing of a general section of \({\mathcal{O}}(\theta)\) on \({\mathbb{P}}({\mathcal{O}}(-\zeta)\oplus{\mathcal{O}}(-2\zeta)\oplus{\mathcal{O}}(-3\zeta))\), where the pushforsward of \({\mathcal{O}}(\theta)\) to \({\mathbb{P}}^2\) is \({\mathcal{O}}(\zeta)\oplus{\mathcal{O}}(2\zeta)\oplus{\mathcal{O}}(3\zeta)\). Then there is a birational map
\(R_3\dashrightarrow U\) over \({\mathbb{P}}^2\), induced by the spinor modification associated to \({\mathcal{F}}\).
Proof. We just need to show that \(\psi_*({\mathcal{F}}^\vee\otimes{\mathcal{F}}) \simeq {\mathcal{O}}\oplus{\mathcal{O}}(\zeta)\oplus{\mathcal{O}}(2\zeta)\oplus{\mathcal{O}}(3\zeta)\). By the computation of
\(\det{\mathcal{F}}\) in the proof of Lemma 23, we have \({\mathcal{F}}^\vee\otimes{\mathcal{F}}\simeq {\mathcal{F}}\otimes{\mathcal{F}}(h-2\zeta)\). Hence, by the long exact sequence obtained by composing the sequence defininfg \({\mathcal{F}}\) with
the ideal sheaf sequence of \(C\subset R_3\), we find: \[\label{eq:big95sequence95spinor95modificatioon95product}
0\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{F}}\otimes{\mathcal{F}}^\vee \longrightarrow{\mathcal{F}}(h-2\zeta)\longrightarrow{\mathcal{F}}\otimes{\mathcal{O}}_C(-2\zeta)\longrightarrow 0.\tag{17}\] We will compute the pushforwards
of all terms one by one.
First term. The pushforward of the first term is identically zero, since \({\mathcal{F}}\) is an abstract spinor bundle.
Third term. With the same argument we used to obtain Equation 17 , we can resolve \({\mathcal{F}}(h-2\zeta)\) as:
\[\label{eq:second95term95equation} 0\longrightarrow{\mathcal{O}}\longrightarrow{\mathcal{F}}(h-2\zeta)
\longrightarrow{\mathcal{O}}(h-2\zeta)\longrightarrow\eta_*{\mathcal{O}}_C(-2\zeta)\longrightarrow 0.\tag{18}\] Here, since \(\psi_*{\mathcal{O}}(h-2\zeta)\simeq{\mathcal{E}}_3(-2\zeta)\), the only nontrivial
term is the last one. However, in light of Lemma 24, we can resolve it as: \[0
\longrightarrow{\mathcal{O}}(-h)\longrightarrow{\mathcal{F}}(-2\zeta)\longrightarrow{\mathcal{O}}(-2\zeta)\longrightarrow\eta_*{\mathcal{O}}_C(-2\zeta) \longrightarrow 0.\] We immediately note that, by projection formula, \(\psi_*{\mathcal{O}}(-2\zeta) \simeq {\mathcal{O}}(-2\zeta)\) and \(\psi_*{\mathcal{F}}(-2\zeta) = 0\). Then, to deal with \(\psi_*{\mathcal{O}}(-h)\), we use
again the Grothendieck duality once we note that \({\mathcal{O}}(-h)\simeq \psi^!{\mathcal{O}}(-3\zeta)[-1]\), and therefore \(\psi_*{\mathcal{O}}(-h)\simeq {\mathcal{O}}(-3\zeta)[-1]\).
Hence: \[{\mathcal{O}}(-3\zeta)[1] = \operatorname{Cone}\left({\mathcal{O}}(-2\zeta)\longrightarrow\psi_*\eta_*{\mathcal{O}}_C(-2\zeta)\right)\] and therefore \(\psi_*\eta_*{\mathcal{O}}_C(-2\zeta) \simeq \bar\eta_*{\mathcal{O}}_L(-2\zeta)\). By splitting the long exact sequence 18 into short exact sequences and applying \(\psi_*\) we get: \[\operatorname{Cone}\left( {\mathcal{O}}\longrightarrow\psi_*{\mathcal{F}}(h-2\zeta) \right) \simeq \operatorname{Cone} \left(
{\mathcal{E}}_3(-2\zeta)\longrightarrow\bar\eta_*{\mathcal{O}}_L(-2\zeta) \right)[-1].\] The right hand side of the last isomorphism, by the first resolution of lemma 24, can be rewritten as \(T(-23\zeta)\oplus{\mathcal{O}}(-\zeta)[-1]\). By a simple application of the Borel–Weil–Bott theorem on \({\mathbb{P}}^2\), we
see that the latter has no higher extensions with \({\mathcal{O}}\), which implies that \(\psi_*{\mathcal{F}}(h-2\zeta)\) is a trivial extension,i.e.: \[\psi_*{\mathcal{F}}(h-2\zeta) \simeq {\mathcal{O}}\oplus T(-3\zeta)\oplus{\mathcal{O}}(-\zeta).\]Fourth term. Since \(|h|\) contracts \(C\) to
a point, the bundle \({\mathcal{F}}(h)\) restricts trivially to \(C\). Hence the fourth term becomes: \[\psi_*{\mathcal{F}}\otimes\eta_*{\mathcal{O}}_C(-2\zeta)
\simeq \bar\eta_*{\mathcal{O}}_L^{\oplus 2}(-2\zeta).\]Summing all up. Putting everything together, we obtain: \[\label{eq:spinor95bundle95proof95better95expression} f_*{\mathcal{F}}\otimes{\mathcal{F}}^\vee \simeq \operatorname{Cone}\left( {\mathcal{O}}(-\zeta)\oplus T(-3\zeta)\oplus{\mathcal{O}}\longrightarrow\bar\eta_*{\mathcal{O}}_L^{\oplus
2}(-2\zeta) \right)[-1].\tag{19}\] Now, let us consider the following diagram: \[\begin{tikzcd} {\mathcal{O}}(-3\zeta)\ar[equals]{d}\ar[hook]{r} & {\mathcal{O}}^{\oplus
2}(-3\zeta)\oplus{\mathcal{O}}(-2\zeta)\oplus{\mathcal{O}}(-\zeta) \ar[hook]{d} & \\ {\mathcal{O}}(-3\zeta) \ar[hook]{r} & {\mathcal{O}}^{\oplus 3}(-2\zeta)\oplus{\mathcal{O}}(-\zeta) \ar[two heads]{d} \ar[two heads]{r} & T(-3\zeta)\oplus
{\mathcal{O}}(-\zeta) \ar[two heads]{d} \\ & \bar\eta_*{\mathcal{O}}_L^{\oplus 2}(-2\zeta) \ar[equals]{r} & \bar\eta_*{\mathcal{O}}_L^{\oplus 2}(-2\zeta) \end{tikzcd}\] where the central row is a direct sum of the Euler sequence with \({\mathcal{O}}(-\zeta)\xrightarrow{\,\,\,I\,\,\,}{\mathcal{O}}(-\zeta)\), and the central column is a direct sum of two copies of the ideal sheaf sequence of \(L\subset{\mathbb{P}}^2\) with \({\mathcal{O}}(-2\zeta)\xrightarrow{\,\,\,I\,\,\,}{\mathcal{O}}(-2\zeta)\). Then, by the snake lemma, the top right corner can be proven to be \({\mathcal{O}}(-3\zeta)\oplus{\mathcal{O}}(-2\zeta)\oplus{\mathcal{O}}(-\zeta)\). Adding \({\mathcal{O}}\xrightarrow{\,\,\,I\,\,\,}{\mathcal{O}}\) to the resulting short exact sequence (right
column) we obtain: \[0\longrightarrow{\mathcal{O}}(-3\zeta)\oplus{\mathcal{O}}(-2\zeta)\oplus{\mathcal{O}}(-\zeta)\oplus{\mathcal{O}}\longrightarrow T(-3\zeta)\oplus {\mathcal{O}}(-\zeta)
\oplus{\mathcal{O}}\longrightarrow\bar\eta_*{\mathcal{O}}_L^{\oplus 2}(-2\zeta) \longrightarrow 0,\] and together with Equation 19 , this concludes the proof. ◻
Corollary 4. Let us call \({\mathcal{B}}(X)\) the Clifford part of the derived category of a conic bundle \(X\). Then, there is a derived equivalence \({\mathcal{B}}(R_3)\simeq {\mathcal{B}}(U)\).
Let \(V\) be a Verra threefold as in Section 4.2. Its anticanonical divisor \(-K_V=-(2H_1+2H_2-3H_1-3H_2)=H_1+H_2\), where \(H_1,H_2\) are the pullbacks of the ample generators of the bases of the two (trivial) \({\mathbb{P}}^2\)-bundle structures. For \(i = 1,2\), let \(f_1, f_2\) be the restrictions of such projections to the Verra threefold. First, we conclude a new proof of Theorem 8,
which removes the genericity assumption for 1-nodal factorial Gushel-Mukai threefold \(X_3\).
New proof of Theorem 8. By Proposition 19 above, there is an equivalence \(\widetilde{{\mathcal{K}}u}(X_3)\simeq{\mathcal{K}}u(V_3)\), Then the equivalence becomes the one \(\Psi:{\mathcal{K}}u(V_3)\simeq{\mathcal{K}}u(V_3')\). Then by [69], the conic bundle \(V_3'\) is the spinor modification of \(V_3\). On the other hand, the discriminant locus of \(V_3\) is a sextic curve on \(\mathbb{P}^2\), so the generic fiber is smooth. Then by [69], \(V_3\)
is birationally equivalent to \(V_3'\). By the birational equivalences \(V_3\sim\widetilde{X_3}\) and \(V_3'\sim\widetilde{X_3'}\)(See [25]), we get \(X_3\sim X_3'\). ◻
Then we apply the same line of thought of Proposition 26: we define a curve \(C\subset V\), and an abstract spinor
bundle on \(V\), which is an extension of the ideal sheaf of \(C\subset V\) with a line bundle. However, as we show in Lemma 27, the associated spinor modification turns out to be trivial.
Lemma 27. Let \(C\) be the intersection of \(V\) with the diagonal \(\Delta\subset{\mathbb{P}}^2\times{\mathbb{P}}^2\). Then one has a short
exact sequence: \[0\longrightarrow{\mathcal{O}}(-2H_1 - H_2)\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{I}}_{C|V}\longrightarrow 0.\] where \({\mathcal{F}}\) is a canonical
abstract spinor bundle, in the sense of [31].
Proof. Call \(\pi_1\) and \(\pi_2\) the projections of \({\mathbb{P}}^1\times{\mathbb{P}}^2\) to the two factors. Denote by \(i: \Delta\xhookrightarrow{\,\,\,\,\,}{\mathbb{P}}^2\times{\mathbb{P}}^2\) the diagonal embedding. Then, the normal bundle of the diagonal is: \[{\mathcal{N}}_{\Delta|{\mathbb{P}}^2\times{\mathbb{P}}^2}\simeq i^*\left( \pi_1^*{\mathcal{O}}(H_1)\otimes\pi_2^*T_{{\mathbb{P}}^2}(-H_2)\right).\] Hence, by the short exact sequence of normal bundles associated to the triple
\(C\subset V \subset {\mathbb{P}}^2\times{\mathbb{P}}^2\), we conclude that \({\mathcal{N}}_{C|V}\simeq j^*(\left( \pi_1^*{\mathcal{O}}(H_1)\otimes\pi_2^*T_{{\mathbb{P}}^2}(-H_2)\right))\).
This allows us to write the Koszul resolution: \[0\longrightarrow{\mathcal{O}}(-2H_1 - H_2)\longrightarrow\Omega^1_{{\mathbb{P}}^2}(-H_1+H_2)|_V \longrightarrow{\mathcal{I}}_{C|V}\longrightarrow 0.\] The proof is concluded
once we observe that \(\Omega^1_{{\mathbb{P}}^2}(-H_1+H_2)|_V\) is the relative cotangent bundle of \(\pi_2\), and hence, up to a twist by a pullback of a line bundle with respect to \(\pi_1^*\), it is isomorphic to the \({\mathcal{F}}^{-1}\) described in [31]. ◻
Thus, it is tempting to conjecture that given any abstract spnor bundle \({\mathcal{F}}\) on \(V\), if the associoated spinor modification is still a Verra threefold, then this bundle
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[69], \(V'\) is the spinor modification of \(V\), then \(V\cong V'\). This would produce an alternative proof of the Categorical Torelli theorem for Verra threefolds.
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It is pointed out by Fei Peng (private communications)↩︎