Projective Hypersurfaces Research Articles

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tib\u{a}r in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

A division theorem for nodal projective hypersurfaces

Let \(V_{n,d}\) be the variety of equations for hypersurfaces of degree d in \({\mathbb {P}}^n({\mathbb {C}})\) with singularities not worse than simple nodes. We prove that the orbit map \(G'=SL_{n+1}({\mathbb {C}}) \rightarrow V_{n,d}\), \(g\mapsto g\cdot s_0\), \(s_0\in V_{n,d}\) is surjective on the rational cohomology if \(n>1\), \(d\ge 3\), and \((n,d)\ne (2,3)\). As a result, the Leray–Serre spectral sequence of the map from \(V_{n,d}\) to the homotopy quotient \((V_{n,d})_{hG'}\) degenerates at \(E_2\), and so does the Leray spectral sequence of the quotient map \(V_{n,d}\rightarrow V_{n,d}/G'\) provided the geometric quotient \(V_{n,d}/G'\) exists. We show that the latter is the case when \(d>n+1\).

On automorphisms of Hilbert squares of smooth hypersurfaces

Let X be a smooth projective hypersurface of dimension at least three. We show that every automorphism of the Hilbert square of X is induced by some automorphism of X.

On cubic fourfolds with an inductive structure

We study the number of planes in four dimensional projective hypersurfaces which have so-called inductive structure. We also determine transcendental lattices of cubic fourfolds of this type.

Zeta functions of projective hypersurfaces with ordinary double points

We extend the approach of Abbott, Kedlaya and Roe to computation of the zeta function of a projective hypersurface with \(\tau \) isolated ordinary double points over a finite field \({\mathbb {F}}_q\) given by the reduction of a homogeneous polynomial \(f \in {\mathbb {Z}}[x_0, \ldots , x_n]\), under the assumption of equisingularity over \({\mathbb {Z}}_q\). The algorithm is based on the results of Dimca and Saito (over the field \({\mathbb {C}}\) of complex numbers) on the pole order spectral sequence in the case of ordinary double points. We give some examples of explicit computations for surfaces in \({\mathbb {P}}^3\).

Systoles and Lagrangians of random complex algebraic hypersurfaces

Let n\geq 1 be an integer, and \mathcal L \subset \mathbb{R}^n be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist c>0 and d_0\geq 1 such that for any d\geq d_0 , any smooth complex projective hypersurface Z in \mathbb{C} P^n of degree d contains at least c\dim H_{*}(Z, \mathbb{R}) disjoint Lagrangian submanifolds diffeomorphic to \mathcal L , where Z is equipped with the restriction of the Fubini–Study symplectic form (Theorem 1.1). If moreover all connected components of \mathcal L have non-vanishing Euler characteristic, which implies that n is odd, the latter Lagrangian submanifolds form an independent family in H_{n-1}(Z, \mathbb{R}) (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For n=2 , the method provides a uniform positive lower bound for the probability that a projective complex curve in \mathbb{C} P^2 of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we provide a similar result for the (n-1) -systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and n tensored by a large power of an ample line bundle over a projective complex n -manifold (Theorem 1.20).

A note on non-uniform points for projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A uniform point for X is a point P such that the projection of X from P has maximal monodromy. We extend and improve some results concerning the finiteness of the locus of non-uniform points for projections of hypersurfaces obtained by the authors and Cuzzucoli (Ann. Mat. Pura ed Appl. 1923, 1–18 (2021)) only for P not contained in X.

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

Equivariant completions of affine spaces

Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства $\mathbb{C}^n$ в полные алгебраические многообразия $X$, для которых действие векторной группы $\mathbb{G}_a^n$ на $\mathbb{C}^n$ параллельными переносами продолжается до действия $\mathbb{G}_a^n$ на $X$. В первой части работы мы подробно изучаем соответствие Хассетта-Чинкеля, описывающее эквивариантные вложения $\mathbb{C}^n$ в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические многообразия и в многообразия Фано определeнных типов. Библиография: 109 названий.

An extension of polar duality of toric varieties and its consequences in Mirror Symmetry

The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called \emph{framed} duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.

Polar degree of hypersurfaces with 1-dimensional singularities

We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the semi-continuity of the polar degree in deformations, and we classify the homaloidal cubic surfaces with 1-dimensional singular locus. Some open questions are pointed out along the way.

A Construction of Multiplicity Class of Hypersurfaces From Hesselink Stratification of a Hilbert Scheme

ABSTRACT It is well known that there is a positive relationship between the maximal multiplicity and the length of associated virtual 1-parameter subgroup of a projective hypersurface. In this article, we will define the multiplicity classes of hypersurfaces and construct them from the Hesselink stratification of a Hilbert scheme.

A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix

We identify the p-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic p as the eigenvalues of a product of special values of a certain matrix of p-adic series. That matrix is a product \(F(\varLambda ^p)^{-1}F(\varLambda )\), where the entries in the matrix \(F(\varLambda )\) are A-hypergeometric series with integral coefficients and \(F(\varLambda )\) is independent of p.

Additive Actions on Toric Projective Hypersurfaces

Let \({{\mathbb {K}}}\) be an algebraically closed field of characteristic zero and \({{\mathbb {G}}}_a\) be the additive group of \({{\mathbb {K}}}\). We say that an irreducible algebraic variety X of dimension n over the field \({{\mathbb {K}}}\) admits an additive action if there is a regular action of the group \({{\mathbb {G}}}_a^n = {{\mathbb {G}}}_a \times \cdots \times {{\mathbb {G}}}_a\) (n times) on X with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action.

Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group S_d. We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

Average size of the automorphism group of smooth projective hypersurfaces over finite fields

In this paper, we show that the average size of the automorphism group over $\mathbb{F}_q$ of a smooth degree $d$ hypersurface in $\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequence of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.

Free rational points on smooth hypersurfaces

Motivated by a recent question of Peyre, we apply the Hardy–Littlewood circle method to count “sufficiently free” rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rationals.

On reconstructing subvarieties from their periods

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.

Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

We show that if a homogeneous polynomial f in n variables has Waring rank $$n+1$$ , then the corresponding projective hypersurface $$f=0$$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of f. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.

Self-duality of the local cohomology of the Jacobian ring and Gherardelli’s theorem

We prove that the 0-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure of Gherardelli’s theorem for the corresponding graded modules. This leads us to a different and characteristic free proof of its self-duality, which turns out to be an easy consequence of Grothendieck’s local duality theorem.