Luis E. Lopez
For a projective hypersurface $X \subset ¶^n$, the images of the polar maps of degree $k$ are studied. The cohomology class defined by these maps is calculated and classical results on dual varieties are presented as applications.
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We use Maxima to compute the moment matrices of hypersurfaces. After that, we compute the Hilbert-Mumford numerical criterion for plane cubics and plane quartics, and give the stability of these curves.
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。
In this paper, we establish a uniqueness theorem for algebraically nondegenerate meromorphic maps of C^m into C P^n and slowly moving hypersurfaces Q_j in C P^n, j=1,...,q in (weakly) general position, where q depends effectively on n and on the degrees d_j of the hypersurfaces Q_j.
One numerical obstruction for rational maps between hypersurfaces
本文针对高维一般超曲面之间的有理支配映射，在附加条件下证明了数值不变量满足 Noether-Fano 型不等式，为有理映射的存在性提供了数值障碍。
We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We then explain a technique of Griffiths which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al.). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weight…
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.
Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.
We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants.
Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.
We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We then explain a technique of Griffiths which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al.). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weight…
Polarized interpolation and normal postulation for curves on Fano hypersurfaces
本文证明Fano超曲面上充分大次数曲线的法丛与余法丛具有良好假设或自然上同调，从而建立极化插值性质。
It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context…
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。
The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to l…
We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.
We give detailed descriptions of the period maps of two 2-parameter families of anti-canonical hypersurfaces in toric 3-folds. One of them is related to a Hilbert modular surface, and the other is related to the product of modular curves.
We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. Lê and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then…
It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a rational map. The group of birational self-maps is either trivial or cyclic of order two.
In this paper, we compute the contributions from double cover maps to genus 0 degree 2 Gromov-Witten invariants of general type projective hypersurfaces. Our results correspond to a generalization of Aspinwall-Morrison formula to general type hypersurfaces in some special cases.
We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.
The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$Φ_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulat…
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.
Automorphisms of hypersurfaces in polarized varieties
本文针对极化簇中的一般超曲面，利用截面环的分次 Betti 数给出统一判据，保证其自同构可延拓至全簇，推广了经典结果至更广簇类。
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 $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. 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 $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…
Relative Polar Multiplicities and the Real Link
本文针对复解析函数定义的超曲面，构造了秩由相对极重数给出的自由阿贝尔群链复形，证明其上同调同构于实链环的约化上同调，并由此建立贝蒂数与相对极重数之间的莫尔斯型不等式。
Non-smoothness of Moduli Spaces of Higher Genus Curves on Low Degree Hypersurfaces
本文研究低次数光滑超曲面上高亏格曲线映射模空间的奇异性，证明次数相对亏格足够大时模空间奇异，并给出奇异轨迹维数下界。
The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. Th…
Let $(C,p_1,\ldots,p_n)$ be a fixed general pointed curve and let $(X,x_1,\ldots,x_n)$ be a smooth hypersurface of degree $e$ and dimension $r$ with $n$ general points. We consider the problem of enumerating maps $f:C\to X$ of degree $d$ (as measured in the ambient projective space) such that $f(p_i)=x_i$. When $e$ is small compared to $r$ and $d$ is large compared to $g$, $e$, and $r$, these numbers have been computed first by passing to a virtual count in Gromov-Witten theory obtained by Buch-Pandharipande, and then by showing (in work of the author with Pandharipande) that the virtual count…
We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.
We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restrict…
In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a pr…
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.
Toric polar maps and characteristic classes
该研究证明复射影空间中超曲面的环面极映射多重度与特定开集的Chern-Schwartz-MacPherson类系数符号一致，给出平面曲线映射次数的局部不变量公式，并构造了任意维数的双有理环面极映射的不可约超曲面族。
Algebraic hyperbolicity of very general hypersurfaces in projective spaces
本文证明了射影空间中非常一般超曲面代数双曲性的最后未解情形，即四次三维簇，完成了分类，方法基于Coskun--Riedl构造及稳定映射模空间和对数几何。
On a Torelli Principle for automorphisms of Klein hypersurfaces
本研究改进Oguiso和Yu的微分方法，给出光滑超曲面自同构的广义三角矩阵表示条件，计算了多类Klein超曲面的自同构群，引入极极化Hodge结构并验证部分情形下自同构群与Torelli原理预测一致。
To every projective variety $X$, we associate a family of hypersurfaces in different Grassmannians, called the coisotropic hypersurfaces of $X$. 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 tangent spaces. We present a new and simplified proof of that result. We show that the coisotropic hypersurfaces of $X$ equal those of its projectively dual variety, and that their degrees are the polar degrees of $X$. Coisotropic hypersurfaces of Segre varieties are defined by hyperdeterminants,…
Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position
本文建立了完备极小曲面环状端上广义高斯映射关于射影簇子一般位置超曲面的修正亏量关系，证明了非退化映射的像不能省略过多超曲面，并首次提出相关的乘积求和不等式，为高斯映射唯一性研究提供了基础。
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison, a perverse sheaf naturally associated to any reduced complex analytic space on which the constant sheaf $\Q_X^\bullet[\dim X]$ is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double point…
Let A be an indecomposable principally polarized abelian variety of dimension g . Third order theta functions embed A in a projective space P(V_3), while second order theta functions embed the Kummer variety K=A/<-1> in a projective space P(V_2). Coble observed that for g = 2 there is a unique cubic hypersurface in P(V_3) that is singular along A, and for g = 3 a unique quartic hypersurface in P(V_2) singular along K . We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group.
In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps from P^1 to P^{N-1} with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar_{0,…
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…
In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify $\Bbb {A}^{n-1}$ into an $(n-1)$-dimensional projective arithmetically Cohen-Macaulay subscheme of some $\Bbb {P}^N$. One particular interesting compactification of $\Bbb {A}^{n-1}$ is the toric variety associated to the Newton polytope of the polynomials defining $f$. We consider two diffe…
In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the Hilbert scheme parameterizing irreducible rational curves of degree $d$ on $X$. We obtain that (1) If $4\leq h\leq n-1$, $R_d(X, h)$ is an integral, local complete intersection of dimension \begin{equation} (n+1-h)d+n-4. \end{equation} (2) If furthermore $(h^2-n)d+h\leq 0$ and $h\geq 4$, in addition to part (1), $R_d(X, h)$ is also rationally connected.
This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating hypergeometric series to solutions of quantum differential equations arising from hypersurfaces in projective space. A particular case of this relationship is a proof of the Mirror prediction for the numbers of rational curves on the Calabi-Yau quintic 3-fold. The approach taken here is entirely algebro-geometric and relies upon a localization formula on the moduli space of stable genus 0 maps to projecti…
On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through $t$ fixed points has general moduli as family of $t$-pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the s…
Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method
本文利用圆法研究低阶超曲面上曲线模空间的奇点，证明了当映射阶数 $e$ 相对亏格 $g$ 足够大时，该模空间至多具有终端奇点。
This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a subm…
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.
A polar hypersurface P of a complex analytic hypersurface germ, f=0, can be investigated by analyzing the invariance of certain Newton polyhedra associated to the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated to f. We develop this general principle of Teissier (see Varietes polaires. I. Invariants polaires des singularites d'hypersurfaces, Invent. Math. 40 (1977), 3, 267-292) when f=0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated to any quasi-ordinary projection of f=0. We build a decomposition of P in bunches…
We are interested in the normal class of an algebraic hypersurface Z of the complex projective space P^n, that is the number of normal lines to Z passing through a generic point of P^n. Thanks to the notion of normal polar, we state a formula for the normal class valid for a general hypersurface Z of P^n. We give a generic result and we illustrate our formula with examples in P^n. We define the orthogonal indidence variety and compute the Schubert class of the variety of projective normal lines to a surface of P^3 in the Show ring of G(1,3). We complete our work with a generalization of Salmon…
In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three point Gromov-Witten invariants up to cubic rat…
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.
An extension of polar duality of toric varieties and its consequences in Mirror Symmetry
本文将 Fano 环面簇的极对偶性扩展为 framed 对偶性，提出一种统一方法以生成任意 Kodaira 维数下环面簇超曲面与完全交集的镜像伙伴，并探讨了其与 Landau-Ginzburg 模型及同调、内蕴镜像对称的联系。
A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality
该研究通过高亏分数的圆法推广，证明了对于一般低次超曲面，不存在增加Fujita不变量的累积映射，从而在特定范围内建立了这些模空间的庞加莱对偶性。
This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations.
We prove a precise version of a general conjecture on the polar degree stated by June Huh. We confirm Huh's conjectural list of all projective hypersurfaces with isolated singularities and polar degree equal to 2.
We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.
Belyi maps from zeroes of hypergeometric polynomials
本文研究超几何多项式零点定义的代数曲面，并应用于参数化亏格为0的Belyi映射族。
The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.
Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This theorem is proved in this paper (Corollary 1.4). The applications of the Multiplicity-Polar theorem are of two types. In the first we construct a deformation so that we can understand the significance of the multiplicity of the pair, or show we can use the multiplicity of the pair to describe some geometric behavior. Applications of this type are the geomet…
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