Marco Andreatta, Elena Chierici, Gianluca Occhetta
Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and either has a fiber type extremal contraction or does not have small extremal contractions b) X has dimension five.
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For a log Fano manifold (X, D) with D\neq 0 and with the log Fano pseudoindex \geq 2, we prove that the restriction homomorphism Pic(X)\to Pic(D_1) of Picard groups is injective for any irreducible component D_1\subset D.The strategy of our proof is to run a certain minimal model program and is similar to the argument of Casagrande's one. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).
The affine cones over Fano-Mukai fourfold of genus $7$ are flexible
本文证明了亏格 $7$ 的任何光滑 Fano-Mukai 四重折叠的仿射锥具有柔性。
Mukai models of Fano varieties
本文给出了Mukai关于指标1且亏格$g \ge 6$的素Fano三维簇分类及其高维推广的自包含简化证明。
We show that the affine cones over any Fano-Mukai fourfold of genus 10 are flexible; in particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano-Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.
We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstrass and Fano or anti-Fano fibrations we are able to describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determ…
Derived categories of Gushel-Mukai surfaces and Fano fourfolds of K3 type
研究K3型Fano四维流形的半正交分解，证明非常一般的对偶Gushel-Mukai表面不相等，尽管导出和L等价。
As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $ρ_X(r_X-1)\geq\dim X-1$ have very special structure, where $ρ_X$ is the Picard number of $X$ and $r_X$ is the index of $X$. In this paper, we classify those $X$ if $ρ_X\leq 3$ or $\dim X\leq 5$.
Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose class [C] belongs to R. We show that s(X)\leq n if n\leq 4. And for n\leq 4, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.
Mukai bundles on Fano threefolds
本文给出 Mukai 定理关于素数 Fano 三维簇上例外向量丛存在的首个完整证明，基于 Lazarsfeld 构造与 K3 曲面理论，支撑双正则分类与半正交分解。
Mukai bundles and Brill-Noether generality for prime Fano threefolds in positive characteristic
针对正特征下亏格≥8的素法诺三维簇，证明其存在对应亏格非平凡分解的唯一向井丛，核心方法是证明其光滑超平面截面为布里尔-诺特一般，丰富了代数几何中相关簇的结构理论。
We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli spac…
The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \math…
Generalizing a question of Mukai, we conjecture that a Fano manifold $X$ with Picard number $ρ_X$ and pseudo-index $ι_X$ satisfies $ρ_X (ι_X-1) \le \dim(X)$. We prove this inequality in several situations: $X$ is a Fano manifold of dimension $\le 4$, $X$ is a toric Fano manifold of dimension $\le 7$ or $X$ is a toric Fano manifold of arbitrary dimension with $ι_X \ge \dim(X)/3+1$. Finally, we offer a new approach to the general case.
Central Weyl involutions on Fano-Mukai fourfolds of genus 10
本文研究了亏格10的Fano-Mukai四重流形上源自G2型Weyl群中心的唯一对合τ，详细刻画了其不动点集合及由τ不变直线扫出的直纹面。
Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
本文研究特殊Gushel-Mukai及相关Fano三维簇的无穷小Torelli问题，证明特殊Gushel-Mukai三维簇的无穷小周期映射不变部分单射，并给出周期映射微分核的几何描述。
The Mukai conjecture for Fano quiver moduli
本文验证了与基本域内部维数向量相关联的 Fano quiver 模空间的 Mukai 猜想。
We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived categories for Enriques and bielliptic surfaces.
This note is an answer to a problem proposed by Ranestad and Iliev. We prove that the projection of general nodal linear sections of suitable dimension of the Mukai varieties $M_g$ are linear sections of $M_{g-1}$.
We construct an explicit complex smooth Fano 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.
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 Lagrangian data; 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 approa…
We perform a systematic study of Gushel-Mukai varieties---quadratic sections of linear sections of cones over the Grassmannian Gr(2,5). This class of varieties includes Clifford general curves of genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieti…
The generalised Mukai conjecture for spherical varieties
本文证明了广义Mukai猜想在有理数因子球面Fano变种中的成立，并引入了衡量Fano变种接近典范程度的最小绝对复杂度项。
We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold with Picard number one, and $E$ is an ample vector bundle of rank two on $X$ with $c_1(X) = c_1(E)$. Equivalently, the present paper completes the classification of ruled Fano $5$-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.
We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very negative line bundle on Y is ample if and only if the bundles parametrized by Y are nef.
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 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement i…
Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
本文证明了Gushel-Mukai簇Kuznetsov分量上Bridgeland稳定性条件的存在性，描述了偶数维情形下的模空间结构，并据此构造了K3型超凯勒簇的新无限系列族及Hodge理论等价判据。
A normal projective variety X is called Fano if a multiple of the anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier divisor H}. Mukai announced, the classification of smooth Fano manifolds X of index i(X)=n-2, under the assumption that the linear system |H| contains a smooth divisor. In this paper we prove that this assumption is always satisfied. Therefore the result of Mukai provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, Mukai manifolds.
Gushel-Mukai varieties
本文综述了Gushel-Mukai簇的几何、模空间、Hodge结构和范畴性质，基于与Kuznetsov的合作及前人研究。
We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conj…
It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}…
Derived categories of generalized Kummer varieties: extended Mukai vector
本文利用扩展 Mukai 向量研究广义 Kummer 型超 Kähler 流形的导出等价，构造并证明了一个在导出等价下不变的整格，并以导出单值群阐述其应用。
We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.
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 that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational…
Effective non-vanishing and Mukai type conjecture
本文引入Fano簇的不变量，提出刻画射影空间乘积的Mukai型猜想，并证明Ambro-Kawamata有效非消失性猜想蕴含该猜想。
We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence is given by a semi-homogenous vector bundle. In particular, given an abelian variety B, and a simple semi-homogenous vector bundle E on B, we will discuss for which E, it can be the image of the structure sheaf of the unit on A under some triangulated equivalence, where A is some abelian variety.
Gushel--Mukai varieties: intermediate Jacobians
该研究刻画了三维与五维Gushel-Mukai簇的中间雅可比簇，证明其同构于关联对偶Eisenbud-Popescu-Walter曲面典范双重覆盖的Albanese簇，并给出了对应簇周期映射的刻画。
We show that the affine cones over a general Fano-Mukai fourfold of genus $g=7$, $8$ and $9$ are flexible. Equivalently, there is an infinitely transitive action of the special automorphism group on such affine cones. In particular, any Mukai fourfold of genus $7,8$ and $9$ is $\Bbb A^2$-cylindrical.
Quadrics on Gushel-Mukai varieties
本文通过将光滑 Gushel-Mukai 簇上的二次超曲面 Hilbert 概型关联到 P^5 点爆破上相关二次超曲面族的线性子空间相对 Hilbert 概型，系统研究了维数 k∈{0,1,2,3} 的二次超曲面分类问题。
We classify pairs $(X,\mathscr E)$ where $X$ is a smooth Fano manifold of dimension $n \geq 5$ and $\mathscr E$ is an ample vector bundle of rank $n-2$ on $X$ with $c_1(\mathscr E) = c_1(X)$.
This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abel…
We compute the Chow groups of smooth Gushel-Mukai varieties of dimension $5$.
A Mukai variety is a Fano n-fold of index n-2. In this paper we study the fundamental divisor of a Mukai variety with at worst log terminal singularities. The main result is a complete classification of log terminal Mukai varieties which have not good divisors, examples of "bad" varieties are given. In such a way we also give a shorter proof of Mukai Conjecture, solved in our previous paper alg-geom/9611024.
Proper Fourier-Mukai partners of abelian varieties and points outside the Fourier-Mukai loci in Matsui spectra
本文通过分析导出范畴的 Matsui 谱，证明了阿贝尔簇的真正 Fourier-Mukai 伙伴仍为阿贝尔簇，并将结果扩展至正常概形；同时揭示了 Fourier-Mukai 轨迹之外的几何结构，为 Ito 猜想提供了基于高维简单阿贝尔簇的新反例。
Algebraic cycles on Gushel-Mukai varieties
研究复 Gushel-Mukai (GM) 簇上的代数环，证明了广义 Hodge 猜想、(动机) Mumford-Tate 猜想和广义 Tate 猜想，计算了除两个无限维情形外的所有积分 Chow 群，并证明了广义伙伴或广义对偶的 GM 簇在中度有理 Chow 动机上同构。
A note on Fourier-Mukai partners of abelian varieties over positive characteristic fields
本文研究正特征域上阿贝尔簇的Fourier-Mukai伙伴，将Krug-Sosna关于Kummer栈的导出等价结果推广到奇特征情形，证明阿贝尔曲面导出等价与Kummer曲面同构等价，并建立超奇异阿贝尔簇的导出Torelli定理。
We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complex…
Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperkähler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., 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 hyperkähler fourfold is in both cases a smooth double cove…
On the deformation theory of Fourier-Mukai transforms between Calabi-Yau varieties
本文研究特征零与混合特征下全忠实 Fourier-Mukai 变换的形变理论，证明源簇具平凡典范丛时形变阻碍可由 Hodge 理论完全控制，推广已有结果并给出阻碍类的上积公式。
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by taking an orthogonal linear section of the spinor tenfold. This is an inverse of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a curve of genus 7 to a Fano threefold of genus 7. An explicit geometric construction of both isomorphisms and a similar result f…
On Mukai's conjecture for hyperelliptic varieties
本文研究数值平凡典范线丛的光滑射影簇的合冲性质，证明了一般性结果，并在任意维数和正特征下验证了阿贝尔簇有限商的Mukai合冲猜想若干情形。
The generalized Mukai conjecture for spherical varieties with a reductive general isotropy group
本文研究具有约化一般迷向群的球面簇，证明了Gagliardi–Hofscheier猜想，并由此推出此类簇上的广义Mukai猜想。
We rework the Mori-Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.
I give three descriptions of the Mukai flop of type $E\_{6,I}$, one in terms of Jordan algebras, one in terms of projective geometry over the octonions, and one in terms of O-blow-ups. Each description shows that it is very similar to certain flops of type $A$. The Mukai flop of type $E\_{6,II}$ is also described.
Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics
本文研究了在正特征和混合特征下阿贝尔簇和K3曲面的傅里叶-穆卡伊伙伴，证明了其提升性质并恢复了原有结果。
We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.
We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.
The Mukai conjecture via Cox rings for smooth canonical ambient embeddings
本文通过Cox环理论研究特殊典范嵌入下的Fano簇，证明了Mukai猜想在局部因子Fano簇中的成立，基于Mori梦想空间的log版本。
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
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