A. Dimca, L. Paunescu
We obtain sharp estimates on the connectivity of complex affine hypersurfaces in terms of the decomposition of the defining equation as a sum of weighted homogeneous components relative to some weight system.
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A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.
We find polyhedral divisors corresponding to the torus action of complexity one on affine trinomial hypersurfaces. Explicit computations for particular classes of such hypersurfaces including Pham-Brieskorn surfaces and rational trinomial hypersurfaces are given.
We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,..…
Counting integral points of affine hypersurfaces
该研究针对仿射超曲面上有界高度的整点计数问题，推导得到了一致上界，推广了Browning、Heath-Brown及作者此前的相关研究结果。
We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simpl…
In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, then we relate first the Jacobian syzygies of $D$ and those of $X$. Then we introduce two types of freeness for an affine hypersurface $X$, and prove various relations between them and the freeness of the projective hypersurface $D$. We write down a proof of the folklore result saying that an affine hypersurface is free if and only if all of its singularities ar…
Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces
本文证明非常仿射超曲面的同调镜像对称性与部分缠绕Fukaya范畴间的辛函子相容，发现补集具有两个由Knörrer周期性关联的自然镜像，并据此修正Auroux猜想，同时提出Liouville流形粘合新方法。
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\C^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th …
We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.
Positivity of polynomials on the nonnegative part of certain affine hypersurfaces
研究多项式在特定仿射超曲面非负部分上的非负性，证明严格正多项式可由正系数多项式表示，推广Pólya定理。
This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.
Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to Petrovski{\uı} and Ole{\uı}nik, Thom and Milnor). Moreover, we show there exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a closed ball in…
We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{C}^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{C}^*$-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X\times X$ and $Y\times Y$.
We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.
In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers $b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we prove an analogous statement for complements of complex affine hyperurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of \cite{FLM,LM06} about $L^2$--Betti numbers of plane curve complements.
Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal rational varieties with torus action of complexity one. For an affine variety X we consider the subgroup SAut(X) of the automorphism group generated by all algebraic subgroups isomorphic to the additive group of the ground field. An affine variety X is rigid if SAut(X) is trivial. In opposite an affine variety is flexible if SAut(X) acts transitively on the regular locus. Arzhantsev proved a criterium for a factorial…
Stable rationality of hypersurfaces in schön affine varieties
本文通过动机方法，将[Nicaise--Ottem'22]的结果从代数环面扩展到schön仿射簇中的超曲面，还应用该方法证明了复Grassmann簇Gr(2,n)中某些超曲面的无理性，与先前研究结果一致。
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then al…
We study the topology of the space of affine hyperplanes $L \subset \CC^n$ which are in general position with respect to a given generic quadratic hypersurface $A$, and calculate the monodromy action of the fundamental group of this space on the relative homology groups $H_*(\CC^n, A \cup L)$ associated with such hyperplanes.
We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb P}^M$, preserving the hypersurface $S$; in particular, for a general hypersurface $S$ the group $\mathop{\rm Aut} ({\mathbb P}^M\setminus S)$ is trivial.
We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwartz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwartz-Halphen's Theorem for Pham-Brieskorn surfaces.
The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.
Weil conjectures and affine hypersurfaces
本文通过将问题简化为超曲面情况，重新证明了有限域上光滑正则簇的黎曼假说，方法结合了变形到仿射超曲面、Artin消失定理及 perverse sheaves 基本性质。
To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $π$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_π\subset{\mathfrak m}$. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for ${\Bbb F}={\Bbb C}$ leads to interesting consequences in singularity theory. Also, for ${\Bbb F}={\Bbb R…
We show that the complement of a degree $d$ hypersurface in a projective complete intersection, whose defining equations have degrees strictly larger than $d$, has a rational connectivity higher than expected. The key new feature is that a positivity result replaces the usual transversality conditions needed to get such connectivity results.
The Russell cubic is a smooth contractible affine complex threefold which is not isomorphic to affine three-space. In previous articles, we discussed the structure of the automorphism group of this variety. Here we review some consequences of this structure and generalize some results to other hypersurfaces which arise as deformations of Koras-Russell threefolds.
It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.
We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in P^{n-1}. In the case when Y is a hyperplane section of a smooth hypersurface in P^{n+1}, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that…
Smooth connectivity in real algebraic varieties
本文提出算法，通过梯度路径计算实代数簇补集的光滑流形的连通分量、欧拉示性数及两点连通性，并应用于奇异点集以确定光滑连通性。
In this paper, we address the following two general problems: given two algebraic varieties in ${\bf C}^n$, find out whether or not they are (1) isomorphic; (2) equivalent under an automorphism of ${\bf C}^n$. Although a complete solution of either of these problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one, and use it to construct numerous examples of isomorphic, but …
Mirror symmetry for very affine hypersurfaces
本文证明光滑拟投影环面 DM 栈边界除子上的凝聚层范畴等价于复环面中超曲面的缠绕 Fukaya 范畴，并通过局部化、拼凑法及新的函子性结果将证明归结为环面簇镜像对称定理。
The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities. We prove that the Jacobian ideal of an affine hypersurfac with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface is of linear type if and only if the corresponding affine curve in the affine chart associated to singular points is locally Eulerian. We prove that the grad…
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…
We use slicing by nongeneric pencils of hypersurfaces and prove a new theorem of Lefschetz type for singular non compact spaces, at the homotopy level. As applications, we derive results on the topology of the fibres of polynomial functions or the topology of complements of hypersurfaces in $\mathbb C^n$.
We prove among other things that a general Calabi-Yau hypersurface in projective space is not rationally swept out by abelian varieties of dimension greater than or equal to 2.
This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in \textsc{Singular}. Among other, we use this algorithm in computational $D$-module theory for computing e. g. the Bernstein-Sato…
We study complements of hypersurfaces in schemes with respect to the property being affine.
The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates isomorphisms of the fundamental groups of the complements to each hypersurface from this family in the p…
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…
An affine algebraic variety $X$ is rigid if the algebra of regular functions ${\mathbb K}[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least 2.
Euler Stratifications of Hypersurface Families
本文根据拓扑 Euler 示性数对射影与甚仿射超曲面族进行分层，提出基于代数几何技术的新算法计算所有分层，并利用临界点计算关系，揭示了 Euler 分层在粒子物理与代数统计中的应用价值。
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.
In this paper we give a description of hypersurfaces with trivial ring $AK(S)$, introduced by the second author as following. Let $X$ be an affine variety and let $G(X)$ be the group generated by all $\Bbb {C}^+$-actions on $X$. Then $AK(X)$ is the subring of all regular $G(X)-$ invariant functions on $X.$ We show that a smooth affine surface $S$ with $AK(S)=\Bbb C$ is quasihomogeneous and so may be obtained from a smooth rational projective surface by deleting a divisor of special form, which is called a ``zigzag''. We denote by $A$ the set of all such surfaces, and by $H$ those which have on…
The Koras-Russell threefold is the hypersurface X of the complex affine four-space defined by the equation x^2y+z^2+t^3+x=0. It is well-known that X is smooth contractible and rational but that it is not algebraically isomorphic to affine three-space. The main result of this article is to show that there exists another hypersurface Y of the affine four-space, which is isomorphic to X as an abstract variety, but such that there exists no algebraic automorphism of the ambient space which restricts to an isomorphism between X and Y. In other words, the two hypersurfaces are inequivalent. The proo…
We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to the case of complete intersections.
In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones.
Koras-Russell threefolds are certain smooth contractible complex hypersurfaces in affine complex four-space which are not algebraically isomorphic to affine three-space. One of the important examples is the cubic Russell threefold, defined by the equation x^2y+z^2+t^3+x=0. In a previous article by A. Dubouloz, P.M. Poloni and the author, the automorphism group of the Russell cubic threefold was studied. It was shown, in particular, that all automorphisms of this hypersurface extend to automorphisms of the ambient space. This has several interesting consequences, including the fact that one can…
We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the co…
We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_τ\subset \mathbb C^n$ with (at worst) isolated singularities. We show that the constancy of the global polar invariants $γ^* (X_τ)$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that $γ^*$-constancy implies C$^\infty$-triviality in the neighbourhood of infinity. We show how the invariants $γ^*$ enter in the description of a CW-complex model of a hypersurface $X_τ$ and therefore provide in particular new invariants at infinity for polyn…
The Hodge structure on the singularity category of a complex hypersurface
本文证明复仿射超曲面奇点范畴的非交换霍奇结构等同于相应射影超曲面的经典霍奇结构，并由此建立霍奇猜想与dg范畴类比之间的等价性。
Let ${\mathcal Q}_n^d$ be the vector space of homogeneous forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $Φ$, introduced in earlier articles by J. Alper, M. Eastwood and the author, that assigns to every form for which the discriminant $Δ$ does not vanish the so-called associated form lying in the space ${\mathcal Q}_n^{n(d-2)*}$. This map is a morphism from the affine variety $X_n^d:=\{f\in{\mathcal Q}_n^d:Δ(f)\ne 0\}$ to the affine space ${\mathcal Q}_n^{n(d-2)*}$. Letting $p$ be the smallest integer for which the product $Δ^pΦ$ extends to a mo…
In this paper we consider the open complement U of a hypersurface Y=V(a) in an affine scheme X. We study the relations between the affineness of U, the intersection of Y with closed subschemes, the property that every closed surface in U is affine, the property that every analytic closed surface is Stein and the superheight of the defining ideal a.
On the Jacobian syzygies for generic toric models
本文针对仿射环面中的一般超曲面，在射影空间关联超曲面配置，并计算了相应雅可比代数的最小分次分解。
This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in P^n. In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the p-adic Picard-Fuchs equation associated with monomial deformations of Fermat hypersurfaces. As a by-product we obtain Poincare Duality for the Rigid cohomology of certa…
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