Some new rational Gushel fourfolds

One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on $\CH_1$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low degree rational curves, parallel to the cubic fourfolds case.

On some families of Gushel-Mukai fourfolds

We give explicit descriptions of some Noether-Lefschetz divisors in the moduli space of Gushel-Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.

On the double EPW sextic associated to a Gushel-Mukai fourfold

In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.

Gushel-Mukai表面的导出范畴与K3型Fano四维流形

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。

Gushel-Mukai 四簇上的圆锥曲线、EPW 六次曲面与 Bridgeland 模空间

Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces

本研究将与极一般Gushel-Mukai四簇相关的两类EPW六次曲面等同于其Kuznetsov分支中稳定对象的Bridgeland模空间，肯定回答了Perry-Pertusi-Zhao的问题，并证明了Kuznetsov-Perry的相关猜想。

Some Moduli of $n$-pointed Fano Fourfolds

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.

Gushel-Mukai 四重簇上由扭曲三次曲线得到的双 EPW 立方体

Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds

本文首次系统研究 GM 四重簇上的扭曲三次曲线，证明双 EPW 立方体是其 Hilbert 概形的 MRC 商，并给出 O'Grady 猜想的新例子。

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 that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational…

Verra 四重体的量子上同调和双有理几何

Quantum cohomology and birational geometry of Verra fourfolds

本文通过计算 Verra 四重体的小量子上同调环，利用原子理论证明了 Verra 四重体不与非常一般的三次四重体或 Gushel-Mukai 四重体双有理等价，并刻画了与之双有理等价的那些四重体的原始上同调结构。

Gushel-Mukai varieties: linear spaces and periods

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…

来自Gushel-Mukai四维簇的Bridgeland模空间的Lagrangian族

Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

本文证明Gushel-Mukai簇与其超平面截面的Kuznetsov分量间拉回与推出函子保持Bridgeland稳定性，并由此构造GM四维簇模空间的8维Lagrangian族及证明GM三维簇模空间的射影性。

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

We show that the stability conditions on the Kuznetsov component of a Gushel-Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel-Mukai fourfolds.

EPW六次曲面与EPW立方体的比较

EPW sextics vs EPW cubes

该研究探讨双重EPW立方体与双重EPW六次曲面的对应关系，通过霍奇结构和椭圆曲线模空间推导族间关联，并证明一般双重EPW立方体是对应Gushel–Mukai四胞体Kuznetsov分量上的稳定对象模空间，解决了Perry等人的问题。

On cylindrical smooth rational Fano fourfolds

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.

特殊Gushel-Mukai流形的K-稳定性

K-stability of special Gushel-Mukai manifolds

本文证明了一般特殊Gushel-Mukai n-重（3≤n≤6）是K-稳定的，并描述了配对(M,cQ)的K-模空间的第一和最后一堵墙，其中M是五次Del Pezzo四重或五重，Q是普通Gushel-Mukai三重或四重。

On the cohomology of Gushel-Mukai sixfolds

We provide a stable rationality construction for some smooth complex Gushel-Mukai varieties of dimension 6. As a consequence, we compute the integral singular cohomology of any smooth Gushel-Mukai sixfold and in particular, show that it is torsion-free.

节点Gushel-Mukai簇的分类消解与双有理几何

Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

本文研究带孤立节点的Gushel-Mukai簇的双有理几何与导出范畴，通过flop建立Kuznetsov分量的分类消解与二次曲面纤维化的Clifford代数模范畴之间的等价，并应用于有理性和双有理模型构造。

K3型法诺四簇

Fano fourfolds of K3 type

本研究从旗流形乘积上齐次向量丛零轨迹构建的法诺四簇数据库中提取64个K3型族，探究其几何与K3结构起源，给出不变量、有理性及导出范畴半正交分解相关信息。

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

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.

Gushel-Mukai 簇上的代数环

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 refinement of some previous results of Bernardara-Marcolli-Tabuada and Ornaghi-Pertusi

The Voevodsky nilpotence conjecture was proved by Bernardara-Marcolli-Tabuada for certain quadric fibrations, intersections of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, and Moishezon manifolds, and later by Ornaghi-Pertusi for certain cubic fourfolds and Gushel-Mukai fourfolds. In this paper we refine these results by using the algebraic equivalence relation instead of the nilpotence equivalence relation. Along the way, we address also certain cases of Küchle fourfolds, families of Sextic del Pezzo surfaces and families of Fano fourfolds of K3 type…

特征 $p\geq 5$ 下偶数维 Gushel-Mukai 簇的 Tate 猜想

The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$

本文研究了特征 $p$ 下维数为 4 和 6 的 Gushel-Mukai 簇，通过借鉴 K3 曲面的证明方法、利用广义伙伴概念及特征 0 下的 Chow 动机同构，证明了特征 $p\geq 5$ 下的 Tate 猜想。

Rational cubic fourfolds in Hassett divisors

We prove that every Hassett's Noether-Lefschetz divisor of special cubic fourfolds contains a union of three codimension-two subvarieties, parametrizing rational cubic fourfolds, in the moduli space of smooth cubic fourfolds.

Bridgeland-Enriques 一般 K3 曲面

Bridgeland-Enriques general K3 surfaces

受 K3 曲面上 Enriques 范畴和 Bridgeland 稳定性条件不变量的启发，引入 Bridgeland-Enriques 一般 K3 曲面的概念，并探讨其与特殊 Gushel-Mukai 三维簇和 Hodge 特殊 Gushel-Mukai 四维簇的联系。

Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds

In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.

由Cremona变换产生的新有理三次四维流形

New rational cubic fourfolds arising from Cremona transformations

本文证明判别式20的非常一般三次四维流形中Fourier-Mukai等价蕴含双有理等价，并借助Veronese曲面定义的Cremona变换发现新的有理三次四维流形。

Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds

We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families $\mathcal C_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.

Potential density of rational points on the variety of lines of a cubic fourfold

We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\mathbb{Z}$.

A very general quartic double fourfold or fivefold is not stably rational

Applying an idea of C. Voisin, we prove that a double cover of P^4 or P^5 branched along a very general quartic hypersurface is not stably rational.

Rational curves of degree 16 on a general heptic fourfold

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a naïve dimension count. In the case of a general hypersurface of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.

Rational cubic fourfolds with associated singular K3 surfaces

Generalizing a recent construction of Yang and Yu, we study to what extent one can intersect Hassett's Noether-Lefschetz divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds $\mathcal{C}$. In particular, we exhibit arithmetic conditions on 20 indexes $d_1,\dots, d_{20}$ that assure that the divisors $\mathcal{C}_{d_1},\dots,\mathcal{C}_{d_{20}}$ all intersect one another. This allows us to produce examples of rational cubic fourfolds with an associated K3 surface with rank 20 Néron-Severi group, i.e. a singular K3 surface.

On the existence of rational curves on projective hyperkähler fourfolds

We show that if $X$ is a projective hyperkähler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a very special case of Oguiso's conjecture for projective hyperkähler fourfolds.

具有辛自同构群的有理三次四重体

Rational cubic fourfolds with a symplectic group of automorphisms

本文证明了具有非2幂阶循环辛自同构群的三次四重体为有理且属于Hassett因子C_d（d=14,42），并描述了相关Lech对结构。

具有素数阶辛自同构的立方四维流形

Cubic fourfolds with a symplectic automorphism of prime order

本文研究了具有素数阶辛自同构的立方四维流形，证明其为理性的条件，并展示了两个非等变理性的有理立方四维流形家族。

Explicit rationality of some cubic fourfolds

Recent results of Hassett, Kuznetsov and others pointed out countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor $C_{14}$ and in [arXiv:1707.00999] for the divisors $C_{26}$ and $C_{38}$. In this note we describe explicit birational maps from a general cubic fourfold in $C_{14}$, in $C_{26}$ and in $C_{38}$ to $\mathbb{P}^4$, prov…

Cubic fourfolds fibered in sextic del Pezzo surfaces

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are rational whenever the fibration has a rational section.

有理纤维化三次四折体

Rational fibered cubic fourfolds

该研究在三次四折体模空间中探究多类已知有理三次除子的交，刻画其不可约分支，给出新的有理del Pezzo曲面或二次曲面纤维化三次四折体显式例子，并研究纤维化有理截面与Brauer类的对应关系。

Rational curves on holomorphic symplectic fourfolds

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral symmetric nondegenerate form, the Beauville form. We give precise conjectures for the structure of the cone of effective curves - and by duality - the cone of ample divisors. Formally they are completely analogous to the results known for K3 surfaces, and they are expressed entirely in terms of the integers represented by the Beauville form restricted to Pic(F…

Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$. We use congruences of 5-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14, 26, 38$ in Hassett's notation.

Some Loci of Rational Cubic Fourfolds

In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus…

Cubic fourfolds containing a plane and a quintic del Pezzo surface

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffi…

关于Gorenstein $\mathbb{Q}_p$-有理三维和四维奇点

On Gorenstein $\mathbb{Q}_p$-rational threefold and fourfold singularities

本文证明对于$n\leq 4$且$p>5$，拟Gorenstein $F$-纯且$\mathbb{Q}_p$-有理$n$维奇点是典范的，基于对对数典范奇点dlt修正的对偶复形的分析。

某些有理椭圆射影四维簇的拓扑刻画与 Hodge 结构

Topological characterization and Hodge structures of some rationally elliptic projective fourfolds

研究嵌入 $\mathbb{P}^8$ 的有理椭圆射影四维簇，借助法丛 Chern 数等刻画其拓扑并证明同构于 $\mathbb P^4$ 的条件，同时验证 Hodge 猜想及非正 Hodge 层级。

Cubic fourfolds containing a plane and K3 surfaces of Picard rank two

We present some families of cubic hypersurfaces in $\mathbb P^5 (\mathbb C)$ containing a plane whose associated quadric bundle does not have a rational section.

A very general quartic double fourfold is not stably rational

We prove that a very general double cover of the projective four-space, ramified in a quartic threefold, is not stably rational.

On a generalization of the Mukai conjecture for Fano fourfolds

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.

Special prime Fano fourfolds of degree 10 and index 2

Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2,5). Special fourfolds correspond to a countable union of hypersurfaces in the …

关于Picard数为一的有理齐次簇上等变向量丛的完全交超凯勒四重簇

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

本文分类了Picard数为一的例外齐次簇上完全可约等变向量丛的零轨迹四重簇，通过霍奇数计算排除超凯勒情形，并确认仅存在Beauville--Donagi和Debarre--Voisin描述的两种超凯勒四重簇。

Balanced Configurations of Lattice Vectors and GKZ-rational Toric Fourfolds in P^6

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R) equivalence and deduce that the only gkz-rational toric four-folds in complex projective space P^6 are those varieties associated with an essential Cayley configuration. In this case, we study a suitable hyperplane arrangement and show that all rational A-hypergeometric functions may be described in terms of toric residues.

Rational curves and bounds on the Picard number of Fano manifolds

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.

On fibrations and measures of irrationality of hyper-Kähler manifolds

We prove some results on the fibers and images of rational maps from a hyper-Kähler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-Kähler fourfold satisfying numerical conditions similar to those considered by O'Grady in his study of fourfolds numerically equivalent to $K3^{[2]}$. We extend his results to this more general context.

复光滑三次三维簇首次稳定的非理性

Irrationality of the first stabilization of a complex smooth cubic threefold

本文证明所有光滑复三次三维簇与射影直线乘积的非理性，并构造一个稳定有理但非理性的 Fano 簇，其高次幂与对称平方均为有理。

通过割线映射研究 Hirzebruch 曲面二重对称积的双有理几何

Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps

本文利用割线与切线构成的 Grassmann 子簇刻画有理正规卷轴长度 2 Hilbert 概形的初等收缩，计算其次数与 $a=1$ 奇点，并阐明 $r=4$ 时相关 Fano 三维簇及 GIT 稳定性。