Gianluca Pacienza
We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf P^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface $X_{d}\subset {\mathbf P}^n$, for $n\geq 6$ and $d\geq 2n-2$, is of general type.
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
Subvarieties of low degree on general hypersurfaces
本文证明了大度数一般超曲面内的小度数子簇均由其与线性空间或其他簇相交生成。
We prove some lower bounds on certain nonegative twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.
Extendable codimension two subvarieties in a general hypersurface
本文在一般超曲面中构造可扩展余维二子簇，并证明当第一陈类远小于次数时，不存在全局生成不可分解 ACM 丛。
A second-order invariant of C. Voisin gives a powerful method for bounding from below the geometric genus of a k-dimensional subvariety of a degree-d hypersurface in complex projective n-space. This work uses the Voisin method to establish a general bound, which lies behind recent results of G. Pacienza and Z. Ran.
We show that for a smooth hypersurface $X\subset \bbP^n$ of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties $Y\subset X$ which are not an intersection $X\cap{S}$ for a codimension two subvariety $S\subset\bbP^n$. We also show there exist $Y\subset X$ as above for which the normal bundle sequence for the inclusion $Y\subset X\subset\bbP^n$ does not split.
We define partially ample subvarieties of projective varieties, generalizing Ottem's work on ample subvarieties, and show their ubiquity. As an application, we obtain a connectedness result for pre-images of subvarieties by morphisms, reminiscent to a problem posed by Fulton-Hansen.
We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead…
We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each example, explicit boundary divisors are also given.
We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Miró-Roig.
The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.
We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
We give the full classification of smooth toric Legendrian subvarieties in projective space. We also prove that under some minor assumptions the group of linear automorphisms preserving given Legendrian subvariety preserves the contact structure of the ambient projective space.
We prove that a general hyperplane section of a smooth Legendrian subvariety in a projective space admits Legendrian embedding into another projective space. This gives numerous new examples of smooth Legendrian subvarieties, some of which have positive Kodaira dimension.
Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\geq codim_{\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\geq[\frac{n}{2}]+1 dimensional quadrics passing through one point.
We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using a relation to the filtration $V$ of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The point is to consider the direct sum of the graded pieces of the multiplier ideals as a module ov…
We consider smooth codimension two subcanonical subvarieties in $\mathbb{P}^n$ with $n \geq 5$, lying on a hypersurface of degree $s$ having a linear subspace of multiplicity $(s-2)$. We prove that such varieties are complete intersections. We also give a little improvement to some earlier results on the non existence of rank two vector bundles on $\mathbb{P}^4$ with small Chern classes.
Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X in $H^0(L^d)$, f deforms in a family such that the corresponding deformations of $Y^r$ dominate $W^r$. Under these hypotheses we give a lower bound for the dimension of a certain linear system on the Cartesian product $Y^r$ having certain vanishing order on a diagonal locus as well as on…
We show that local complete intersection subvarieties of smooth projective varieties, which have partially ample normal bundle, possess the G2-property. This generalizes results of Hartshorne and Bădescu-Schneider.
We define the characteristic numbers of a holomorphic k-distribution of any dimension on $mathbb P^n$ and obtain relations between these numbers and the characteristic numbers of an invariant subvariety. As an application we bound the degree of a smooth invariant hypersurface.
We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (up to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.
We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3 contains lines but no other rational curves.
Algebraic hyperbolicity of subvarieties of homogeneous varieties
本文将超曲面的代数双曲性结果推广到高余维完全交，证明在射影空间中非常一般的完全交在次数和满足特定阈值条件时具有或不具有代数双曲性。
Low degree subvarieties of universal hypersurfaces
本文研究了通用超曲面的低度子变种，证明了在大度数条件下，子变种与参数化的射影变种交集的关系，并展示了大度数假设的必要性。
We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of rational homogeneous varieties are complete intersections.
Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I d…
On a conjecture on Hodge loci of linear combinations of linear subvarieties
本文针对 Movasati 关于包含相交 $k$-平面的三次和四次超曲面 Hodge 轨迹维数的猜想，在 $k \geq 5$ 时给出反例，并在 $k=3,4$ 时提供支持该猜想的新证据。
A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a consequence, we obtain a non-trivial description of the value of a Kloosterman-sum-like exponential sum.
We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.
I prove that every smooth legendrian variety generated by quadrics is a homogeneous variety and further I give a list of all such legendrian varieties. A review of the subject is included, illustrated by examples. Another result is that no complete intersection is a legendrian variety.
I will present an explicit formula for the intersection indices of the Chern classes of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion--Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an…
Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq λd^{ε_1}+c(\sum_{1\leq h < ε_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ o…
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
On potential density of integral points on the complement of some subvarieties in the projective space
本文研究了数域上 $n$ 维射影空间中特定闭子簇余集上整点的潜在密度，推广了 Corvaja-Zucconi 在低维射影空间中的相关结果。
We work over an algebraically closed field of arbitrary characteristic. Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong limitations. Their whole argument is derived from the fact that the sectional genus of surfaces of degree d lying on a hypersurface of degree s varies in an interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a similar result with an interval whose length depends only on s. The main point is Lemma 1.1 whose proof is a direct application of the posi…
When a reductive group $G$ acts linearly on a complex projective scheme $X$ there is a stratification of $X$ into $G$-invariant locally closed subschemes, with an open stratum $X^{ss}$ formed by the semistable points in the sense of Mumford's geometric invariant theory which has a categorical quotient $X^{ss} \to X//G$. In this article we describe a method for constructing quotients of the unstable strata. As an application, we construct moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data (an '$n$-rigidification') on a projective base.
A subvariety of a complex projective space has a well-known dual variety, which is the set of its tangent hyperplanes. The purpose of this paper is to generalise this notion for a subvariety of a quite general partial flag variety. A similar biduality theorem is proved, and the dual varieties of Schubert varieties are described.
This paper is a report based on the results obtained during a three months internship at the University of Pittsburgh by the first author and under the mentorship of the second author. The notion of an amoeba of a subvariety in a torus $(\mathbb{C}^*)^n$ has been extended to subvarieties of the general linear group $GL_n(\mathbb{C})$ by the second author and Manon. In this paper, we show some basic properties of these matrix amoebas, e.g. any such amoeba is closed and the connected components of its complement are convex when the variety is a hypersurface. We also extend the notion of Ronkin f…
Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibr…
A short way of counting maps to hypersurfaces in Grassmannians
该研究利用Quot概形紧化计算从亏格g光滑射影曲线到格拉斯曼流形超曲面的d次映射虚计数，并探究大次数d下该虚计数在合适超曲面条件下是否具有枚举性。
To each subvariety $X$ in projective $n$-space of codimension $m$ we associate an integer sequence of length $m + 1$ from $1$ to the degree of $X$ recording the maximal cardinalities of finite, reduced intersections of $X$ with linear subvarieties. We call this the sequence of secant indices of $X$. Similar numbers have been studied independently with the aim of classifying subvarieties with extremal secant spaces. Our focus in this note is the study of the combinatorial properties that the secant indices satisfy collectively. We show these sequences are strictly increasing for nondegenerate s…
This paper continues the study of non-general type subvarieties begun in a joint paper with M.Schneider and A.Sommese (Int.L.Math. 10, 1999). We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and moreover study the rational quotient. We also relate the fundamental groups and prove a cohomological criterion for a manifold to be rationally connected.
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 ver…
We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one contained in a $\Bbb P^{N-h-1}$. We prove that one of the following happens: (i) there exists an integer $r$, $r<N-n$ such that $X$ is contained in a variety $V_r$ of dimension at most $N-r$ containing a family of dimension $h+1$ of subvarieties of dimension $N-h-r$, each one contained in a linear space of dimension $N-h-1$; (ii) The degree of $Y$ is bounded by a fu…
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant 2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we prove that if $X$ is general, then for any $p\in X$ and $2 \leqslant h\leqslant \min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when $X$ is a very general hypersurface of degree $d\geqslant 2n+2$, we describe the relation between the cones $V^h_p$ and the degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general …
We show that if $X$ is a smooth uniruled projective variety and $L$ a big and semiample $\mathbb{Q}$-divisor on $X$, then there exists a proper closed subset $W\subset X$ such that every subvariety $Y$ satisfying $a(Y,L)> a(X,L)$ is contained in $W$.
Variational Hodge conjecture for complete intersections on hypersurfaces in projective space
该研究针对射影空间中超曲面上完全交闭链的变分Hodge猜想，提出了一种全新的简化证明方法，为相关代数几何问题的研究提供了新的论证思路。
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.
In this paper, we discuss classical derivation of the residue integral representation of the d=1 rational Gromov-Witten invariants of projective hypersurfaces that followed from localization technique.
In this paper we present some properties for projective hypersurfaces, smooth and singular, to be criteria for identification. To make the decision with these criteria, we have included procedures written in Singular language.
This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-Kähler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.
Some rational subvarieties of moduli spaces of stable vector bundles
本文研究了在模空间中具有固定行列式和秩的稳定向量丛族，通过格拉斯曼变种参数化，构造了与之双有理的特殊子簇。
We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result).
The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective $3$-space.
Hodge conjecture for projective hypersurface
本文通过证明霍奇类为解析对数De Rham类，并证明此类为代数闭链类，从而证明了光滑射影超曲面的霍奇猜想。
This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincaré series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different configurations of isolated singularities. As a by-product of a recent result of E. Sernesi, we give examples of classical hypersurfaces which are (or are not) projectively rigid. We also include a Singular program to compute the invariants and to decide if a singular projective hypersurface is nodal and projectively rigid.
We study Gauss-Manin systems of non tame Laurent polynomial functions. We focuse on Hori-Vafa models, which are the expected mirror partners of the small quantum cohomology of smooth hypersurfaces in weighted projective spaces.
We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that there are only finitely many families of quasi smooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension.
Automorphisms of weighted projective hypersurfaces
本文研究拟光滑复加权射影超曲面的自同构群，证明自同构在多数情形下可延拓到外围空间，给出线性自同构群有限的刻画及一致上界，并描述了给定权和次数的通用超曲面的自同构。
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。