Mukai Varieties Research Articles

Kuznetsov’s Fano threefold conjecture via K3 categories and enhanced group actions

Abstract We settle the last open case of Kuznetsov’s conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel–Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel–Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.

Chow groups of Gushel–Mukai fivefolds

Abstract We compute the Chow groups of smooth Gushel–Mukai varieties of dimension 5.

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperk\"{a}hler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

On the automorphisms of Mukai varieties

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for \(7\leqslant g\leqslant 10\). In the Appendix by Y. Prokhorov, the latter statement is extended to \(g=12\).

K3 curves with index k>1

Let mathcal {KC}_g ^k be the moduli stack of pairs (S, C) with S a K3 surface and Csubseteq S a genus g curve with divisibility k in mathrm {Pic}(S). In this article we study the forgetful map c_g^k:(S,C) mapsto C from mathcal {KC}_g ^k to {mathcal {M}}_g for k>1. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when S is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending C in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether c_g^k dominates the locus in {mathcal {M}}_g of k-spin curves with the appropriate number of independent sections. We are able to do this only when S is a complete intersection, and obtain in these cases some classification results for spin curves.

Gushel–Mukai varieties with many symmetries and an explicit irrational Gushel–Mukai threefold

We construct an explicit smooth Fano complex threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel–Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action. Along the way, we construct Gushel–Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud–Popescu–Walter sextic with a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action discovered by the second author in 2013.

Gushel--Mukai varieties: intermediate Jacobians

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds.

On Morin Configurations of Higher Length

AbstractThis paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length—a question naturally related to the theory of Gushel–Mukai varieties. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the canonical genus $6$ curve $C_{\ell }$ union of the $10$ lines in a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. More in general an irreducible family of special configurations of length $\geq 11$, we name as Morin–Del Pezzo configurations, is considered and studied. This includes the configuration of maximal cardinality and families of configurations of lenght $\geq 16$, previously unknown. It depends on $9$ moduli and is defined via the family of nodal and rational canonical curves of $Y$. The special relations between Morin–Del Pezzo configurations and the geometry of special threefolds, like the Igusa quartic or its dual Segre primal, are focused.

Gushel–Mukai varieties: Moduli

We describe the moduli stack of Gushel–Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of so-called Lagrangian data defined in our previous works; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel–Mukai varieties and construct some complete nonisotrivial families of smooth Gushel–Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli stack.

Gushel–Mukai varieties: Linear spaces and periods

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic 4-fold and the primitive second cohomology of its variety of lines, a smooth hyper-Kähler 4-fold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel–Mukai varieties of dimension 4 (resp., 6), that is, smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp., of the cone over Gr(2,5) and a quadric). The associated hyper-Kähler 4-fold is in both cases a smooth double cover of a hypersurface in P5 called an Eisenbud–Popescu–Walter sextic.

Projections of Mukai varieties

This note is an answer to a problem proposed by Iliev and Ranestad. We prove that the projections of general nodal linear sections of suitable dimension of Mukai varieties $M_g$ are linear sections of $M_{g-1}$.

Derived categories of Gushel–Mukai varieties

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

Degenerations to unobstructed Fano Stanley–Reisner schemes

We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley–Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. Furthermore, we show that the Stanley–Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial $$T^2$$ . This can be used to find new toric degenerations of linear sections of $$G(2,n)$$ .

Scrollar syzygies of general canonical curves with genus $\le 8$

We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in these cases. We have generalized results 1 and 3 to general curves of even genus in math.AG/0108078. Result 2 is the main new result of this paper.

Classification of varieties with canonical curve section via gaussian maps on canonical curves

The purpose of this article is to develop further a method to classify varieties X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] having canonical curve section, using Gaussian map computations. In a previous article we applied these techniques to classify prime Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index and Mukai varieties , i.e., varieties of dimension four or more with canonical curve sections. First we determine when the Hilbert scheme [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] of such varieties X is nonempty. Moreover, in the case of Picard number one, we prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is irreducible and that the examples of Fano-Iskovskih and Mukai form a dense open subset of smooth points of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].