We study real nonsingular projective cubic fourfolds up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in the middle homology. Moreover, we provide a graph whose vertices represent the equivalence classes of such cubics and edges represent their adjacency. It turns out that this graph essentially coincides with the graph characterizing a certain adjacency of real non-polarized K3-surfaces.
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…
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.
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.
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…
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.
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.
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.
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…
Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any ε>0, we show that X contains O(B^{D+ε}) rational points of height at most B. The implied constant in this estimate depends upon the choice of εand the coefficients of the cubic form defining X.
Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.
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…
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.
Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms that share no common variables. ----- This paper features an appendix "Groupe de Brauer non ramifié des hypersurfaces cubiques singulières (d'après P. Salberger)", by J.-L. Colliot-Thĺène.
Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k. We show that the equation g=0 has infinitely many integer solutions when the cubic part of g defines a projective hypersurface with singular locus of dimension <n-10. The proof is based on the Hardy-Littlewood circle method.
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
Mathematical School of Science, Xiamen University（厦门大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
There are easy "polynomial" deformations of Calabi-Yau hypersurfaces in toric varieties performed by changing the coefficients of the defining polynomial of the hypersurface. In this paper, we explicitly constructed the ``non-polynomial'' deformations of Calabi-Yau hypersurfaces, which arise from deformations of the ambient toric variety.
In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using invariant theory, we use Shamash's construction with a cone over the spinor variety. We also describe an algebro-geometric interpretation of our matrix factorization which connects the spinor tenfold and the Cartan cubic.
In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation …
Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined étale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in $P^n$.
We prove that for any cubic polynomial of slice rank $r$, the intersection of all linear subspaces of minimal codimension contained in the corresponding hypersurface has codimension $\le r^2+\frac{(r+1)^2}{4}+r$ in the affine space. This is deduced from the following result of independent interest. Consider the intersection $I$ of linear ideals $(P_i)$ in $k[x_1,\ldots,x_n]$, with $\dim P_i\le r$. Then the number of quadratic generators of $I$ is $\le r^2$.
The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are identified using subalgebras of a deformed Jacobian ring.
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.
In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of Hypergeometric series on certain $CM$ points. Our methods is based on the calculation of the Picard-Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.
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.
According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how to respond to the chirality question: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in…
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…
We give a way to construct group of pseudo-automorphisms of rational varieties of any dimension that fix pointwise the image of a cubic hypersurface of $P^n. These group are free products of involutions, and most of their elements have dynamical degree >1. Moreover, the Picard group of the varieties obtained is not big, if the dimension is at least 3. We also answer a question of E. Bedford on the existence of birational maps of the plane that cannot be lifted to automorphisms of dynamical degree >1, even if we compose them with an automorphism of the plane.
We give a geometric derivation of Schottky's equation in genus four for the period matrices of Riemann surfaces among all period matrices. The equation arises naturally from the singularity theory of the Gauss map on the theta divisor, and thus generalizes for any genus $g \ge 4$ to a certain ideal of Siegel modular forms vanishing on period matrices of Riemann surfaces. This ideal is generated by modular forms associated to the invariants of cubic forms in $g - 1$ variables which vanish on the Fermat cubic.
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 are interested in obstructions to the FIRST order deformation of a pair of a smooth hypersurface $f_0$ and a smooth curve $C_0$ contained in $f_0$. In the first half of the paper, we give necessary conditions for the pair to deform in the first order. In particular, for a rational curve $C_0$, this necessary condition is $$H^1(N_{C_0}f_0(1))=0.$$ In the second half, we apply the necessary conditions from the first half of the paper to study the geometry of smooth curves in hypersurfaces. The main application is for the case where $C_0$ is a rational curve.
Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on…
