János Kollár
Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.
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Unirationality and geometric unirationality for hypersurfaces in positive characteristics
本文基于Segre和Kollár的三次超曲面研究，构造正特征不完全域上的特定超曲面，证明正则且有理点Zariski稠密的几何有理概形未必单有理，特征2的部分三次曲面也有此特性。
Unirationality of hypersurfaces via highly tangent lines
本文提出基于高切线的射影空间低次完全交单有理性构造，应用于超曲面时得到更优界：当n≥2^((d-1)2^(d-5))时，射影n空间中一般d≥6次超曲面单有理，改进经典结果。
Let $\mathscr{X}\to W$ be a flat family of generically irreducible hypersurfaces of degree $d\geq 2$ in $\PP^n$ with singular locus of dimension $t$, with $W$ unirational of dimension $r$. We prove that if $n$ is large enough with respect to $d$, $r$ and $t$, then $\mathscr{X}$ is unirational. This extends results in \cite {Pre, HMP}.
We give examples of smooth $\k$-unirational line-free quartic hypersurfaces over a non algebraically closed field $\k$. Unlike other methods of proving unirationality, our method does not rely on existence of linear spaces on quartics.
The aim of this short note is to define the \it universal cubic fourfold \rm over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett divisors $\mathcal{C}_d$, in the range $8\leq d \leq 42$. By applying inductively this argument, we are able to show that, in the same range of values, $\mathcal{C}_{d,n}$ is unirational for all integer values of $n$. Finally, we observe that for explicit infinitely many values of $d$, the universal cubic fourfold over $\mathcal{C}_d$ can not be unirational.
We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.
We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and minimal cubic surfaces admit parameterizations of degree 6. It is an open question whether or not minimal cubic surfaces over finite fields can admit degree 3 or 4 parameterizations.
Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.
Equivariant Unirationality of Cubic Threefolds
本文利用Cassels--Swinnerton-Dyer猜想的证明，完成复三次三维簇的等变单有理分类，证明Klein三次三维簇的PSL2(F11)-单有理性和本质维数为3，反驳了Dolgachev的猜想。
A transfer principle for unirationality
本文利用多项式集体强度建立单有理性的转移原理，证明高次级强度多项式对应单有理超曲面，并推出 Fano 概形及完全交普遍性的相关结论。
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 provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.
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.
We study rational surfaces on very general Fano hypersurfaces in $\mathbb{P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree $d$ sufficiently close to $n$ and $n$ sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio $α$, a very general hypersurface in $\mathbb{P}^n$ of degree $d$ sufficiently close to $n$ will admit no maps fr…
The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomo…
We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other …
The purpose of this note is to prove Grothendieck's standard conjectures for the Fano variety of lines on a smooth cubic hypersurface in projective space.
We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.
Closed points on cubic hypersurfaces
本研究推广Coray关于三次超曲面上闭点的相关结果，证明三次超曲面的特定对称积具有稳定双有理性质。
We study the connected component of the automorphism group of a cubic hypersurface over complex numbers. When the cubic hypersurface has nonzero Hessian, this group is usually small. But there are examples with unusually large automorphism groups: the secants of Severi varieties. Can we characterize them by the property of having unusually large automorphism groups? We study this question from the viewpoint of prolongations of the Lie algebras. Our result characterizes the secants of Severi varieties, among cubic hypersurfaces with nonzero Hessian and smooth singular locus, in terms of prolong…
Degenerations to secant cubic hypersurfaces and limiting Hodge structure
本文研究光滑三次超曲面族退化到割线型三次超曲面时的极限混合Hodge结构，并利用Usui的部分紧化刻画周期映射的局部延拓。
We show that if a cubic hypersurface with positive dual defect over the complex number field is not a cone, then either the hypersurface coincides with the secant variety of the singular locus, or the hypersurface contains a linear subvariety of dimension greater than the dual defect such that the intersection of the singular locus and a general contact locus is contained in the linear subvariety.
Rational curves on cubic hypersurfaces in positive characteristic
研究特征不等于2、3的三次超曲面上有理曲线的模空间，证明当超曲面维度大于等于4时，Kontsevich模空间是不可约的。
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.
We prove weak approximation for smooth cubic hypersurfaces of dimension at least 2 defined over the function field of a complex curve.
The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in $\mathbb{P}^4$, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect in two …
We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its ${\rm CH}_0$-group. We prove that for odd dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition. For cubic threefolds $X$, this turns out to be equivalent to the algebraicity of the minimal class $θ^4/4!$ of the intermediate Jacobian $J(X)$. In dimension $4$, we show that a special cubic fourfold with disc…
In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients.
We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.
In this paper we classify three-dimensional singular cubic hypersurfaces with an action of a finite group $G$, which are not $G$-rational, are not $G$-birationally isomorphic to a quadric and have no birational structure of $G$-Mori fiber space with the base of positive dimension. Also we prove the $A_{5}$ -birational superrigidity of the Segre cubic.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
Linear spaces in Hessian loci of cubic hypersurfaces
本文研究立方超曲面的Hessian领域，证明当维度超出预期时，立方形式必为Thom-Sebastiani类型，并分析投影线性空间的存在性。最终证明唯一与Waring秩6立方形式具有相同Hessian的光滑三次曲面是自身，且任意超曲面的Hessian非锥面。
Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $Λ$ of dimension $h\geq \max\{2,\dim(Λ\cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then $X_4$ is unirational over $k$. This improves previous results by A. Predonzan and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a density result for the $k$-rational points of quartic $3$-folds with a double plane over…
By an additive structure on a hypersurface S in projective space we mean an effective action of commutative unipotent group on projective space which leaves S invariant and acts on S with an open orbit. It is known that these structures correspond to pairs (R,H) of local finite-dimensional algebra R and a hyperplane H in the maximal ideal of R. We show when a projective hypersurface of degree 3 has an additive structure and when structure is unique.
A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at leas…
Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of the Cartan cubic, in finitely many ways. To each such "Cartan representation" we associate a rank nine vector bundle on $X$ with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov. Finally we show that the generic eight dimensional section of t…
Algebraic Gromov's ellipticity of cubic hypersurfaces
本文证明了 P^{n+1}（n>1）中的光滑三次超曲面在 Gromov 意义下具有代数椭圆性，首次给出了此类非有理射影流形的实例，并推导出其穿孔仿射锥也具有椭圆性。
In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Comput…
If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant of the hessian matrix of $f$. Hypersurfaces with vanishing hessian were studied systematically for the first time in the fundamental paper [GN], where Gordan and M. Noether analyze Hesse's claims in [Hesse1, Hesse2] according to which these hypersurfaces are necessarily cones. Of course cones have vanishing hessian. Clearly the claim is true if deg(X)=2 so th…
Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
研究退化到特定割线型奇异三次曲面的三次超曲面变形的极限霍奇结构，此前已研究过相关退化问题，此次继续探讨塞维里簇超平面截面的极限霍奇结构，其能联系簇的几何与霍奇理论退化。
Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space
研究立方超曲面的奇点性质，并通过线性代数方法描述射影空间射影化切丛的超平面截面，扩展了Mazouni-Nagaraj的高维结果。
On the Hessian of cubic hypersurfaces
研究一般三次超曲面的 Hessian 轨迹，描述其奇异轨迹与消歧，并证明一般三次四维簇的 Hessian 超曲面奇点为一般类型极小曲面。
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.
Toute hypersurface cubique lisse complexe de dimension au moins 2 dont l'équation est donnée par l'annulation d'une somme de formes cubiques à variables séparées, chaque forme impliquant au plus trois variables, est universellement $CH_{0}$-triviale. --- If a smooth cubic hypersurface of dimension at least 2 is defined by the vanishing of a sum of cubic forms in independent variables and each of these forms involves at most 3 variables, then the cubic hypersurface is universally $CH_{0}$-trivial : there is an integral Chow decomposition of the diagonal.
The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension dimension. In particular, we thus rec…
Motivic obstruction to rationality of a general cubic hypersurface in $\mathbb P^5$
研究$\mathbb{P}^5$中一般三次超曲面合理性问题，引入整本质不可分解动机，证明相关动机性质及提升定理，为将三次四维形猜想归结为正特征下算术现象提供可能。
Cylinder maps of algebraic cycles on cubic hypersurfaces
本文证明含一次一维环的光滑三次超曲面的直线簇与超曲面的Chow群柱映射满射，并给出三次四维簇直线簇上积分Hodge猜想的新证明及积分Tate猜想的确认。
In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurfaces $X$. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape's theorem that $\mathrm{CH}_1(X)$ is always generated by lines and that it is isomorphic to $\Z$ if the dimension of $X$ is at least 5. Another application is to the intermediate jacobian of a cubic threefold $X$. To be more precise, we show that the intermediate jacobian of $X$ is naturally isomorphic to the Prym-Tjurin variety constructed from …
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…
We show that every morphism from a degree 5 hypersurface in 4-dimensional projective space to a nonsingular degree 3 hypersurface in 4-dimensional projective space is necessarily constant. In the process, we also classify morphisms from the projective plane to nonsingular cubic threefolds given by degree 3 polynomials.
We show that smooth cubic hypersurfaces of dimension $n$ defined over a finite field ${\bf F}_q$ contain a line defined over ${\bf F}_q$ in each of the following cases: - $n=3$ and $q\ge 11$; - $n=4$ and $q\ne 3$; - $n\ge 5$. For a smooth cubic threefold $X$, the variety of lines contained in $X$ is a smooth projective surface $F(X)$ for which the Tate conjecture holds, and we obtain information about the Picard number of $F(X)$ and its 5-dimensional principally polarized Albanese variety $A(F(X))$.
We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces
本文利用锥形消解及其推广形式，并结合紧 GIT 商的显式描述，计算节点三次、四次平面曲线及三次曲面方程空间与模空间的有理上同调。
In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of P^{n+1}, for a fixed n > 1. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p<2^{n+1} and that any such hypersurface admitting an automorphism of order p>2^n is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli…
We study a double cover $ψ:X\to V\subset\mathbb{P}^{n}$ branched over a smooth divisor $R\subset V$ such that $R$ is cut on $V$ by a hypersurface of degree $2(n-\mathrm{deg}(V))$, where $n\geqslant 8$ and $V$ is a smooth hypersurface of degree 3 or 4. We prove that $X$ is nonrational and birationally superrigid.
This paper designs an alogrithm to compute the minimal combinations of finite sets in Euclidean spaces, and applys the algorithm of study the moment maps and geometric invariant stability of hypersurfaces. The classical example of cubic curves is repeated by the algorithm. Furhtermore the alogrithm works for cubic surfaces. For given affinely indepdent subsets of monomials, the algorithm can output the unique unstable points of the Morse strata if it exists. Also there is a discussion on the affinely dependent sets of monomials.
A note on special cubic fourfolds of discriminant 14 and non-minimal K3 surfaces of degree 10
本文研究判别式14的特殊三次四折Hassett除子中的一般三次四折，证明其包含特定的10次非极小K3曲面，明确了该曲面的线构成与二次超曲面包含关系。
We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold P^1\timesP^2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.
Rational approximation on cubic hypersurfaces
针对数域上存在k-有理直线的光滑三次超曲面，研究其有理点的逼近常数计算问题，完成了对应常数的推导并验证了Coba猜想，为相关代数几何与数论研究提供了支撑。
We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…
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