Radu Laza
We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
We study the lattices of algebraic and transcendental cycles of cubic fourfolds.
We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.
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 give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperkahler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold non containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we reprove the categorical version of Torelli Theorem for cubic fourfolds, we obtain the identification of the period point of LLSvS eightfold with that of the Fano variety, and we discuss derived Torelli Theorem for cubic fourfolds.
In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using Bridgeland stability conditions. Secondly, we compare automorphism groups of cubic fourfolds with automorphism groups of their associated K3 surfaces. Thirdly, we note that the existence of a non-trivial symplectic automorphism on a cubic fourfold is related to the existence of associated K3 surfaces.
Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds
本文研究了Peskine六重形与关联的K3表面及立方四重形的关系，证明了特定条件下立方四重形的Fano线族与Debarre-Voisin超kähler四重形的同构性。
We list all finite abelian groups which act effectively on smooth cubic fourfolds.
In the first part of this paper we will prove the Voevodsky's nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing a $τ$-plane $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.
We study the number of planes for four dimensional projective hypersurfaces which has so-called inductive structure. We also determine transcendental lattices for cubic fourfolds of this type.
We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.
We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of intersection of Hassett divisors. We show that Fermat cubic fourfold is contained in every Hassett divisor, which yields a new proof of Hassett's existence theorem of special cubic fourfolds. We obtain an algorithm to determine the irreducible components of the intersection of any two Hassett divisors and we give new examples of rational cubic fourfolds. More…
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-nece…
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.
Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.
We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.
In this note, we give a short proof of the Torelli theorem for cubic fourfolds that relies on the global Torelli theorem for irreducible holomorphic symplectic varieties proved by Verbitsky.
We prove that cubic fourfolds in a certain 10-dimensional family have finite-dimensional motive. The proof is based on the van Geemen-Izadi construction of an algebraic Kuga-Satake correspondence for these cubic fourfolds, combined with Voisin's method of "spread". Some consequences are given.
We study lines and twisted cubics on cubic fourfolds with simple isolated singularities. We show that the Hilbert scheme compactification of the total space of Starr's fibration on the space of twisted cubics on a cubic fourfold with simple isolated singularities and not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight. The latter admits a crepant resolution which is a deformation of the symplectic eightfold constructed from the space of twisted cubics on a smooth cubic fourfold. Thereby we give another proof of the fact that the latter is …
Cubic fourfolds of discriminant 24 and rationality
本文研究判别式24的三次四重簇，通过其扭曲曲面中Brauer类消失的条件，证明了有理性质，并构造了可数无穷多个新的有理例子。
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.
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。
Cubic fourfolds containing highly singular hyperplane sections
本研究构造了五个不可约除子，用于参数化含高度奇异超平面截面的光滑立方四重面，并证明这些除子不是Noether-Lefschetz除子。
A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic $2$-cycles on them, it is natural to ask for the most algebraic cubic fourfolds $X$ to which this conjecture is still applicable. In this paper, we show that for an appropriate `algebraicity index' $κ_X$, there exists a unique class of cubics maximizing $κ_X$, not having an associated $K3$ surface; namely, the cubic fourfolds…
The equations of general Hassett maximal cubic fourfolds
本文研究Hassett极大立方四维流形，构造了最大维数为十六的locus的不可约分量，并利用代数和算术方法分析相关晶格。
On the irrationality of cubic fourfolds
该研究证明了立方四重曲面的原始上同调群与K3曲面的中间上同调群同构，揭示了其无理性特征。
We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga's compactification associated to this arrangement. This work builds on and is a natural continuation of our previous paper on the GIT compactification of the moduli space of cubic fourfolds.
There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, we show a cubic fourfold with a particular anti-symplectic involution has an associated K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hasset…
Fourier-Mukai numbers of K3 categories of very general special cubic fourfolds
本文针对非常一般特殊三次四维流形，研究其K3范畴的Fourier-Mukai伙伴数量，并给出了相应的计数公式。
In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperkähler varieties, such as the Fano variety of lines and the LLSvS 8fold.
This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and their geometric interpretations, and connections with rationality and unirationality constructions.
We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.
Cubic fourfolds with symplectic automorphisms
本文通过分类特定群的表示，确定具有辛自同构的立方四维流形的射影方程及模空间的不可约分量数量，并探讨了其定义域。
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。
Equivariant birational geometry of cubic fourfolds and derived categories
本文从导出范畴角度研究等变双有理性，构造了有限群在光滑三次四维流形上非线性化但稳定线性化的作用实例。
Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.
We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.
It is proved that the non-rationality of a generic cubic fourfold follows from a conjecture on the non-decomposability in the direct sum of non-trivial polarized Hodge structures of the polarized Hodge structure on transcendental cycles on a projective surface.
We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.
In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on $r$ parameters, and in particular they are of wild representation type; special surfaces on the cubic fourfolds are explicitly constructed by Macaulay2; a new $19$-dimensional family of projective ten-dimensional irreducible holomorphic symplectic manifolds associated to a certain cubic fo…
L-equivalence and Fourier--Mukai partners of cubic fourfolds
本文研究了三次四维流形的L等价与傅里叶-穆卡伙伴，通过衍生Torelli型准则和判别式条件，证明了计数公式并展示了有限的L等价类。
In this paper, we develop a new method to classify abelian automorphism groups of hypersurfaces. We use this method to classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. A parallel result (Theorem 5.1) is obtained for quartic surfaces.
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.
We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.
Lines on cubic threefolds and fourfolds containing a plane
该研究刻画了特征非2域上含平面的一般三次三维簇的直线Fano概形，以此刻画非闭域上这类三次三维簇的有理性，并显式构造含平面三次四维簇直线Fano簇的拉格朗日纤维化。
We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number $m$ of Fourier-Mukai partners of its associated K3 surface, if $d \equiv 2 (\text{mod}\,6)$; else, if $d \equiv 0 (\text{mod}\,6)$, the cubic fourfold has $\lceil m/2 \rceil$ Fourier-Mukai partners.
Relative compactified Prym and Picard fibrations associated to very good cubic fourfolds
本文证明极好三次四重对应的相对紧化Prym与Picard纤维化的规范化是光滑的，推广了Laza、Saccà和Voisin的结果。
The Fano variety of lines on singular cyclic cubic fourfolds
本文研究奇异循环三次四重簇上直线Fano簇的辛解消，特别关注沿具有$A_i$型孤立奇点的三次三重簇分支的情形，并发现覆盖自同构诱导的三阶非辛自同构。
We study the Brill-Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill-Noether special associated K3 surface of degree 14. As an application, the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is contained in the irreducible locus of cubic fourfolds containing two disjoint p…
Planes in cubic fourfolds
该研究确定了复光滑三次四折体中平面的最大数量为405（仅费马三次四折体可达）、实光滑三次四折体中实平面的最大数量为357（由Clebsch–Segre三次四折体实现），且射影等价下平面数超350的三次四折体仅有三类。
We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.
An error in Section 4 invalidates all the main results of the paper.
Non-symplectic automorphisms of prime order of O'Grady's tenfolds and cubic fourfolds
本文分类 OG10 型不可约全纯辛流形的素数阶非辛自同构，刻画其由三次四维流形诱导的情形，并用于探讨相关有理性猜想。
Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds
本文证明了非常一般的三次四维流形在L等价时必为同构，同时指出存在特殊三次四维流形在L等价但非同构。当三次四维流形在特定Hassett除子中非常一般时，L等价的流形也是傅里叶-莫克伴侣。此外，作者还提供了进一步的例子支持L等价的超Kähler流形应为D等价的猜想。
A note on cubic fourfolds containing several planes
研究包含多个平面（特别是沿直线相交的两个平面）的四次三维簇的几何、Hodge理论和导出范畴，并关联到扭曲K3曲面。
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 $\mathbb{Z}$.
In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold X in the divisor and using point counting methods over finite fields to show X is Noether-Lefschetz general. We also show that two other divisors considered in [ibid.] are not Noether-Lefschetz divisors.
Cubic fourfolds, Kuznetsov components and Chow motives
本文证明了具有Fourier-Mukai导出等价Kuznetsov分支的两个光滑三次超曲面四维簇，其Chow动机作为Frobenius代数对象同构，并推得l-adic上同调Frobenius代数间存在Galois等变同构，同时讨论了与K3曲面导出等价的情形。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。