E. Carlini, L. Chiantini, A. V. Geramita
We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\PP^n$. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.
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We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize to them and the tropicalization of their intersection.
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。
Automorphism groups of rigid complete intersections
研究射影空间中次数严格递增的超曲面完全交的自同构群，利用组合刚性条件证明其自同构群是定义超曲面自同构群的交集，并应用此原理于两个超曲面完全交的自然族，确定了相关自同构群。
C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.
We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
Rank varieties over the generic hypersurface I
本文研究了在通用超曲面上的模和复形的秩品种，通过扩展标量而非限制标量的方法定义，证明了任何投影代数簇均可作为有限生成模的秩品种，并探讨了这些秩品种的性质。
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
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 $\le r^2$.
We study the intersection of two particular Fermat hypersurfaces in $\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its quotients. Explicit computation of the quotients is done using a Gröbner basis algorithm. We also study the $p$-rank, zeta function, and number of rational points, of the modulo $p$ reduction of the curve. We show that the Jacobian of the genus 2 quotient is $(4,4)$-split.
Intersections of Real Symmetric Hypersurfaces
本文证明了贝祖定理的对称版本，表明复对称超曲面横截相交的轨道类型由次数决定，并在射影平面和三维复射影空间中分类了交轨迹的轨道类型，由此给出实对称曲线交集中实点数量的强限制。
On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.
We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space $G/H$. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of $G/H$ admits a description in terms of volumes of polytopes.
On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel is known since one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in a $n$ dimensional complex space? This problem is solved in this work in the case $n=3$. As an application, the correlation functions of bosonic string theories are constructed on a canonical curve of genus four.
Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results are amoeba analogs of Bernstein's Theorem and Bézout's Theorem providing an upper bound for the number of connected compo…
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…
We give upper bounds for the dimension of the set of hypersurfaces of $\mathbb{P}^N$ whose intersection with a fixed integral projective variety is not integral. Our upper bounds are optimal. As an application, we construct, when possible, hypersurfaces whose intersections with all the varieties of a family of integral projective varieties are integral. The degree of the hypersurfaces we construct is explicit. ----- On majore la dimension de l'ensemble des hypersurfaces de $\mathbb{P}^N$ dont l'intersection avec une variété projective intègre fixée n'est pas intègre. Les majorations obtenues s…
Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces
研究r个线性无关d次超曲面在射影空间中零维交集的最大可能大小，提出精确公式猜想并在m=2时证明，进而计算射影Reed-Muller码的广义汉明重量。
In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called intersection tree by intersection theory. We construct a sequence of i…
A classical result of Boole shows that, in characteristic 0, the set of singular degree d hypersurfaces in P^N is a divisor of degree (N+1)(d-1)^N in the projective space of all hypersurfaces. We give here analogous formulae for complete intersections in P^N of arbitrary codimension and degrees, in any characteristic. ----- Un resultat classique de Boole montre que, sur un corps de caracteristique 0, l'ensemble des hypersurfaces singulieres de degre d dans P^N est un diviseur de degre (N+1)(d-1)^N de l'espace projectif de toutes les hypersurfaces. On obtient ici des formules analogues pour des…
Let $W\subset \mathbb {P}^n$, $n\ge 3$, be a degree $k$ hypersurface. Consider a "general" reducible, but connected, curve $Y\subset \mathbb {P}^n$, for instance a sufficiently general connected and nodal union of lines with $p_a(Y)=0$, i.e. a tree of lines. We study the Hilbert function of the set $Y\cap W$ with cardinality $k°(Y)$ and prove when it is the expected one. We give complete classification of the exceptions for $k=2$ and for $n=k=3$. We apply these results and tools to the case in which $Y$ is a smooth curve with $\mathcal {O}_Y(1)$ non-special.
Matrix factorizations of a hypersurface yield a description of the asymptotic structure of minimal free resolutions over the hypersurface. We introduce a new concept of matrix factorizations for complete intersections that allows us to describe the asymptotic structure of minimal free resolutions over complete intersections.
We consider the closed locus of $r$-tuples of hypersurfaces in $\mathbb{P}^r$ with positive dimensional intersection, and show in a large range of degrees that its largest component is the locus of $r$-tuples of hypersurfaces whose intersection contains a line. We then apply our methods to obtain new results on the largest components of the locus of hypersurfaces with positive dimensional singular loci and the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.
In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb{P}^5$. We thus prove a stronger Bogomolov-Gieseker inequality for characters of stable vector bundles and stable objects on $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.
On the diagonal of quartic hypersurfaces and $(2,3)$-complete intersection $n$-folds
研究非常一般的四次超曲面和(2,3)-完全交集n-折叠的对角线分解问题，利用循环理论技术与Voisin的结果，扩展了稳定有理性到对角线分解存在的结论，并给出新证明。
Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$
本文通过计算加权射影空间超曲面的第二上同调相交形式并描述低次整系数上同调群，为识别奇异 Fano 簇光滑化得到的 Fano 流形提供了显式工具。
In this work we study algebraic, geometric and topological properties of the Milnor classes of local complete intersections with arbitrary singularities. We describe first the Milnor class of the intersection of a finite number of hypersurfaces, under certain conditions of transversality, in terms of the Milnor classes of the hypersurfaces. Using this description we obtain a Parusiński-Pragacz type formula, an Aluffi type formula and a description of the Milnor class of the local complete intersection in terms of the global Lê cycles of the hypersurfaces that define it. We consider next the ge…
Generalized Hypergeometric Functions for Degree k Hypersurface in CP^{N-1} and Intersection Numbers of Moduli Space of Quasimaps from CP^{1} with Two Marked Points to CP^{N-1}
本文将 CP^{N-1} 中 k 次超曲面的镜像计算所用广义超几何函数实现为带两个标记点的拟映射模空间相交数的生成函数，建立了超曲面镜像对称与拟映射计数之间的具体联系。
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersur…
Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third.
Let $V$ be a degree $d$, reduced hypersurface in $\mathbb{CP}^{n+1}$, $n \geq 1$, and fix a generic hyperplane, $H$. Denote by $\mathcal{U}$ the (affine) hypersurface complement, $\mathbb{CP}^{n+1}- V \cup H$, and let $\mathcal{U}^c$ be the infinite cyclic covering of $\mathcal{U}$ corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules $H_i(\mathcal{U}^c;\mathbb{Q})$ of the hypersurface complement and show that, if $i \leq n$, these are torsion over the ring of rational Laurent polynomials. We also…
Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in t…
Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $CP^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $CP^1$ with Two Marked Points to $CP^{N-1}$
本文将 $CP^{N-1}$ 中 Calabi-Yau 超曲面镜像计算的周期积分导出为从带两标记点的 $CP^1$ 到 $CP^{N-1}$ 的拟映射模空间相交数的生成函数。
Special points on intersections of hypersurfaces
本文建立超曲面交线在Arnold-Shimura意义下具有稠密“level ℓ”点集的维数下界，方法基于Gómez-Gonzáles-Wolfson的可解点框架，并推广到其他辅助无理数类，以理解“特殊点”的算术性质。
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…
The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove that for a Fano complete intersection $X\subset \mathbb{P}^n$ of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on $X$ has the expected dimension.
We establish a link between the GIT stability of complete intersections of same degree hypersurfaces in the same ambient projective space, which can be parametrised as tuples in a Grassmanian scheme, and the log canonical thresholds of hypersurfaces contained in these tuples. We also establish a similar link in the case of Variations of GIT quotients, where we consider tuples of complete intersections and distinct hyperplanes.
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics…
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected.
We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials…
We show that the dual of the Cayley cone, associated to a Minkowski sum decomposition of a reflexive polytope, contains a reflexive polytope admitting a nef-partition. This nef-partition corresponds to a Calabi-Yau complete intersection in a Gorenstein Fano toric variety degenerating to an ample Calabi-Yau hypersurface in another Fano toric variety. Using the Batyrev-Borisov mirror symmetry construction, we found the mirror contraction of a Calabi-Yau complete intersection to the mirror of the ample Calabi-Yau hypersurface.
The quadratic Euler characteristic of a smooth projective same-degree complete intersection
本文提出一种算法，用于计算同次超曲面构成的光滑射影完全交的二次欧拉示性数，并以两个广义Fermat超曲面的完全交为例进行了计算。
We generalize Werner's defect formula for nodal hypersurfaces in $\mathbb P^{4}$ to the case of a nodal complete intersection threefold.
We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed as a complete intersection of ample hypersurfaces in a smooth toric variety. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric MMP, we prove that in each case, $H_2(X,\mathbb{Z})$ is generated by classes of rational curves.
In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the $\mathbb{R}$-degree of related sets such as the boundary of amoebas and the amoeba of the real part of hypersurfaces defined over $\mathbb{R}$. For all these objects, we provide bounds for the respective $\mathbb{R}$-degrees.
The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety. We focus on the hypersurface case. Let $M$ be a complex manifold, $X\subset M$ a singular hypersurface. We study residues of top-dimensional meromorphic forms with poles along $X$. Applying resolution of singularities sometimes we are able to construct residue classes either in $L^2$-cohomology of $X$ or in the intersection cohomology. The conditions allowing to construct these classes coincide. They can be formulated in terms of the weight filtration. …
We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface $X\subset \bbP^{4}$ of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in $\bbP^4$. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.
In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair $(X,Z)$ where $X$ is a smooth complete intersection in the projective space and and $Z$ is a simple normal crossing divisor in $X$ whose irreducible components are smooth hypersurface sections on $X$. Our Jacobian rings give an algebraic description of the cohomology of the open complement $X-Z$ and it is a natural generalization of the Poincaré residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize the Macaulay's duality the…
Framed duality and mirror symmetry for toric complete intersections
本文将完全环面簇中的超曲面族$f$-镜像对称系统扩展至完全交子簇族，基于分次对偶发展了相关理论，并探讨了广义Landau-Ginzburg/完全交对应。
We show that the Quantum Lefschetz Hyperplane Principle can fail for certain orbifold hypersurfaces and complete intersections. It can fail even for orbifold hypersurfaces defined by a section of an ample line bundle.
Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface
本文证明了 Edoukou-Ling-Xing 猜想在三次超曲面情形（d=3，q≥7）下成立，即非退化 Hermitian 簇与三次超曲面的最大交点数仅由 d 个特殊超平面并集达到。
We prove that a complete intersection of $c$ very general hypersurfaces of degree at least two in $N$-dimensional complex projective space is not ruled (and therefore not rational) provided that the sum of the degrees of the hypersurfaces is at least $\tfrac{2}{3} N + c + 1$. To this end we consider a degeneration to positive characteristic, following Kollár. Our argument does not require a resolution of the singularities of the special fiber of the degeneration. It relies on a generalization of Kollár's "algebraic Morse lemma" that controls the dimensions of the second-order Thom-Boardman sin…
Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.
The classical version of Bézout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bézout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bézout's Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of Bézout's The…
In this paper we propose and prove an explicit formula for computing the degree of Fano schemes of linear subspaces on general hypersurfaces. The method used here is based on the localization theorem and Bott's residue formula in equivariant intersection theory.
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely t…
Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective l…
Lifts of cycles in tropical hypersurfaces and the Gamma conjecture
本文研究了热带超曲面中环的提升，并通过热带交理论计算其交数，进而推导了与镜像对称Gamma猜想相关的周期积分渐进行为。
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