Pierrette Cassou-Noguès, Daniel Daigle
We determine the Newton trees of the rational polynomials of simple type, thus filling a gap in the proof of the classification of these polynomials given by Neumann and Norbury.
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Following Contou-Carrere [CC], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. We prove that the rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called, the combinatorial tangent space. For this, we use a characterization of rational smoothness of a Schubert variety introduced by Carrell and Peterson [CP].
We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric surface whose torus fixpoints are either smooth or cyclic T-singularities (in the sense of Wahl) of type $\frac{1}{r^2}(1, kr - 1)$. We also show that any exceptional sequence can be transformed by mutation into an exceptional sequence which consists only of objects of rank one.
We classify (semi)stable sheaves on a rational curve with one node. The results are based on the classification of indecomposable torsion-free sheaves due to Drozd and Greuel "Tame and wild projective curves and classification of vector bundles", where the sheaves are described in terms of certain combinatorial data. We translate the condition of (semi)stability into this combinatorial language and solve the so obtained problem.
Combinatorial structure of low degree rational curves on a smooth Hermitian surface
本文研究了光滑赫米特曲面中低度有理曲线的组合结构，揭示其与强正则图和关联方案的关系。
Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic a topological type (cubic, points), and combinatorial pencil the cyclic sequence of five combinatorial reducible cubics. Up to the action of the symmetric group $S_5$ on $\{2, ...6\}$, there are seven possible combinatorial pencils with node at 1. Consider now the set of six pencils obtained, making the node to be $1, ...6$. Up to the action of S_6 on ${1, …
On Stable Rationality of Polytopes
本文证明一大类多面体在包含关系下保持非稳定有理性，并据此提出组合策略，改进了特征 0 下射影空间非稳定有理超曲面的界。
We give a simple proof of the statement that every rational curve in the primitive class of a general K3 surface is nodal.
A T-variety is an algebraic variety X with an effective regular action of an algebraic torus T. Altmann and Hausen gave a combinatorial description of an affine T-variety X by means of polyhedral divisors. In this paper we compute the higher direct images of the structure sheaf of a desingularization of X in terms of this combinatorial data. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a partial criterion for a T-variety to be Cohen-Macaulay. As an application we characterize in this terms quasihomogeneous elliptic singularities of su…
The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.
We give an algorithm for calculating the splitting type of the normal bundle of any rational monomial curve. The algorithm is obtained by reducing the calculus to a combinatorial problem and then by solving this problem.
In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi follows from the results of M. Borodzik and C. Livingston in the case of rational cuspidal curves with two critical points.
Combinatorial Hodge Index Theorem for Polytopes
该研究将Maxim-Schuermann公式从顶点有理的多面体推广至任意凸多面体，依托BBFK的组合相交上同调框架，还探讨了多面体的霍奇指标定理。
Rational Normal Curves, Chip Firing and Free Resolutions
本文通过筹码点火博弈与 parcycle 图的组合关联研究有理正规曲线，为每个 Cohen-Macaulay 初始单项式理想显式构造 Gröbner 退化及其极小自由消解，并给出胞腔消解、Hilbert 级数公式和 Eagon-Northcott 复形的组合解释。
We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before.
A combinatorial proof of the $λ_g$ conjecture in genus 2
本研究通过将$\lambda_2$类表示为边界层的线性组合，结合投影公式、弦方程和胀子方程，用归纳法给出了亏格2下$\lambda_g$猜想的简洁组合证明。
For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant schemes along an open subset of a particular irreducible component.
Combinatorial Cycle Classes in the Intersection Cohomology of Projective Toric Varieties
本文探讨了极小循环类在极小扇形交集上同调中的表现，验证了在投影典范变种中，极小循环类生成偶次交集上同调空间。
We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric $g$ and $h$ polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two com…
Combinatorial proofs of the type A quiver component formulas
本文用纯组合方法替代Gröbner退化与几何技巧，给出A型箭图轨迹的K理论及上同调箭图分量公式的新证明。
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