Carlos D'Andrea, Amit Khetan
We present an explicit formula for computing toric residues as a quotient of two determinants, a la Macaulay, where the numerator is a minor of the denominator. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.
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We provide a formula to compute the big Cohen-Macaulay test ideal for triples $((R,Δ),\mathfrak{a}^{t})$ where $R$ is a mixed characteristic toric ring and $\mathfrak{a}$ is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero.
Tropical computations for toric intersection theory in Macaulay2
本文介绍了Macaulay2包TropicalToric.m2，该包利用tropical几何进行环面相交理论计算。
We give a purely algebraic proof of the hypersurface case of Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.
LipschitzSaturation: A Macaulay2 Package for Computing Lipschitz Saturations of Modules and Toric Varieties
介绍用于Macaulay2的LipschitzSaturation软件包，它能计算模和环面簇的Lipschitz饱和。在模设置中处理多种饱和概念，通过基于曲线的成员测试解决计算难题，还实现了环面奇点构造算法，该软件包免费且需特定版本的Macaulay2。
Our main result states that for a toric surface in $P^4$ canonical module is always Cohen-Macaulay.
We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties.
Building on our earlier work on toric residues and reduction, we give a proof for the mixed toric residue conejecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory toric quotients and the other in the calculus of toric residues. This quickly leads to the proof of the conjecture, which gives a closed form for the summation of a generating series whose coefficients are certain variants of the "numbers" of rational curves on toric complete int…
We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotie…
In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or arithmetically Cohen-Macaulay (Gorenstein) and having a Cohen-Macaulay tangent cone or a local ring with non-decreasing Hilbert function, from just one single example with the same property, verifying Rossi's conjecture for larger classes and extending some results appeared in literature.
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of $f_1,...,f_r$. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
Consider an n-dimensional projective toric variety X defined by a convex lattice polytope P. David Cox introduced the toric residue map given by a collection of n+1 divisors Z_0,...,Z_n on X. In the case when the Z_i are T-invariant divisors whose sum is X\T the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope P to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals I of the homogeneous coordinate ring of X. We give a necessary and suf…
In ${\bf C}^{n+1}$, one can show that the residue of $n+1$ homogeneous forms of the same degree equals the integral of a certain $(n,n)$ form over ${\bf P}^n$. Furthermore, the Jacobian of the forms has nonzero residue equal to a certain intersection number. In this paper, we generalize these results to an arbitrary projective toric variety. In particular, we define a toric version of the Jacobian and show that it has the correct properties, and we give various integral formulas for the toric residue. We also review some commutative algebra results of Batyrev and Danilov and discuss the relati…
We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.
We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a…
Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formul…
The paper consist of two parts. In the first part we introduce flags of lattices and associated injective systems of (non-normal) cones and projective systems of (non-normal) affine toric varieties. We study the associated field of multidimensional Laurent power series. In the second part we use the resolution of singularities techniques to study the geometry near a complete flag of subvarieties and the Parshin's residues. The first part plays the role of a standard coordinate neighborhood for Parshin's points.
The toric residue is a map depending on n+1 semi-ample divisors on a complete toric variety of dimension n. It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to s…
On Bott's residue formula for toric complete intersections
本文研究了紧致纹理轨道上完全交集的博特残差公式，通过分析正则分布和建立庞加莱型不等式，探讨了分布奇点数目的计算方法及其应用。
ADE Classification of Hypersurface Singularities over Local Rings
本文基于Cohen-Macaulay型对完全正则局部环上的超曲面进行ADE型分类，并将有限与可数Cohen-Macaulay型的经典结果推广到依赖于剩余域基数的更大基数情形。
In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in Macaulay2. We showed in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Botbol-09] we generalized this to the case where the base locus is not necessarily a local complete intersection. The key point consists in exploiting the sparse structu…
Terminal 3-folds that are not Cohen-Macaulay
本文证明在特征 $p$（$p=2,3,5$）或混合特征 $(0,p)$ 下，3 维终端奇点可非 Cohen-Macaulay，通过正则概形关于 $p$ 阶循环群的商构造，并给出仅具环面奇点的充分条件。
Counting aCM Toric Bundles of Rank Two on the Veronese Surface
本文通过 Klyachko 滤过方法，对 Veronese 曲面上环面等变秩二 aCM 向量丛的同构类（模超平面类扭转）进行计数，并给出代表性例子。
We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max Nöther's $AF+BG$ Theorem, Macaulay's Theorem, and Kollár's Effective Nullstellensatz, as well as recent results by Hickel and Andersson-Götmark.
Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobi's Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity', in projective or toric situations.
We present a new integration formula for intersection numbers on toric quotients, extending the results of Witten, Jeffrey and Kirwan on localization. Our work was motivated by the Toric Residue Mirror Conjecture of Batyrev and Materov; as an application of our integration formula, we obtain a proof of this conjecture in a generalized setting.
Toric Euler-Jacobi vanishing theorem and zeros at infinity
本文研究了托里安欧拉-雅各比消失定理与无穷远处零点存在的关系，通过几何条件和商环维数分析，建立了全局消失与零点存在的等价性。
Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular characteristic class, the Chern-Schwartz-MacPherson class, of a complete simplicial toric variety X defined by a fan from the combinatorial data contained in the fan. Specifically, we give an effective combinatorial algorithm to compute the Chern-Schwartz-MacPherson class of X, in the Chow ring (or rational Chow ring) of X. This method is formulated by combining, a…
We use localization to describe the restriction map from equivariant Chow cohomology to ordinary Chow cohomology for complete toric varieties in terms of piecewise polynomial functions and Minkowski weights. We compute examples showing that this map is not surjective in general, and that its kernel is not always generated in degree one. We prove a localization formula for mixed volumes of lattice polytopes and, more generally, a Bott residue formula for toric vector bundles.
For a class of monadic deformations of the tangent bundles over nef-Fano smooth projective toric varieties, we study the correlators using quantum sheaf cohomology. We prove a summation formula for the correlators, confirming a conjecture by McOrist and Melnikov in physics literature. This generalizes the Szenes-Vergne proof of Toric Residue Mirror Conjecture for hypersurfaces.
Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case
本文研究超椭圆曲线在特征为2的残差域下的聚类、托里克秩和2-秩性质，通过定义相对稳定模型并分析其特殊纤维的图结构，给出计算方法及阈值判定。
We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with isolated cyclic quotient singularities {S_i} admits a Q-Gorenstein partial smoothing to a surface with singularities given by the residues of the S_i. We define the singularity content of a Fano lattice polygon P: this records the total number of elementary T-singularities and the residues of the corresponding toric Fano surface X_P. We express the degree of X_…
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