Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudfoot, Botong Wang
We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering the main result of [AHK18]. We also show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring of M.
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Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincare duality for matroid Chow rings.
Inequalities for Chow Polynomials and Chern Numbers of Matroids
本文研究Matroid的Chow多项式和Chern数的不等式，通过概率分布解释归一化Chow系数，并推导出关于平坦标志数量和Chow多项式根的不等式，同时利用Hirzebruch χ_y-特征类建立新的Matroid Chern数不等式。
Chow classes of matroids and standard Young tableaux
本文通过研究蛇形矩阵的Chow类，将之与特定的ribbon Schur函数关联，并提供了Schubert基中系数的显式公式，同时简化了经典结果并证明了某些Schubert系数的正性。
Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. Dotsenko conjectured that the Chow ring of any matroid is Koszul. The purpose of this paper is to prove Dotsenko's conjecture. We also show that the augmented Chow ring of a matroid is Koszul. As a corollary, we show that the Chow rings a…
Chow polynomials and chain polynomials of matroids are not log-concave
该研究构造显式无限族拟阵，证明其Chow、链等多项式不满足对数凹性，否定了相关实零点猜想，还给出图结合体h多项式对数凹性的反例。
Schubert coefficients of sparse paving matroids
本文将Grassmannian中Chow类在Schubert基下展开的系数定义为拟阵的Schubert系数，计算了非连通拟阵及所有稀疏铺砌拟阵的该系数，并证实了其非负性。
Decompositions of Chow rings of direct sums of matroids
本文研究拟阵直和的周环结构，证明了周环与增广周环的对偶递归模分解，导出欧拉数新递归公式，推广了已有结论并得到弱秩偏序集周多项式的类似分解。
The $\mathbb{S}_n$-equivariant Chow polynomial of the braid matroid
本文确定了辫形矩阵B_n的S_n等变Chow多项式的生成函数，通过将其与多尺度微分的模空间联系起来，贡献于多尺度微分模空间的拓扑研究。
The singular cohomology ring of a uniform matroid: combinatorics and Lefschetz properties
本文研究均匀拟阵奇异上同调环的组合结构与Lefschetz性质，通过Koszul同调构造显式基，推导Hodge数公式并证明其满足拟射影强Lefschetz性质。
Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
本文研究了拟阵Chow理论与Kazhdan-Lusztig框架下四个对象的Hilbert级数，发展了多项式与环级数间的平行关系，利用赋值特性提出快速计算方法，并证明了均匀拟阵的实根性及多项式系数最大化等猜想，将单峰性强化为$\gamma$-正性。
Simplicial generation of Chow rings of matroids
本文提出拟阵 Chow 环的单纯生成元表示，借助拟阵商解释建立单项式基与相对 Schubert 拟阵的双射，并证明体积多项式对数凹及 1 次 Hodge–Riemann 关系，且无需 AHK18 的翻转技术。
Tautological classes of matroids
本文通过引入拟阵的重言类这一几何框架，统一并推广了拟阵理论与代数几何交叉领域的最新成果，建立了Tutte多项式四变量变换的Chow理论描述、对数凹性及例外Hirzebruch-Riemann-Roch型公式。
Augmented singular cohomology, uniform matroids, and real-rootedness
本文研究均匀拟阵相关环面簇的奇异上同调环，构造显式基，证明修改后的扇满足强Lefschetz性质，并证明精细Hodge--Poincaré多项式的实根性，解决猜想并推广已知定理。
Intersection theory of matroids: variations on a theme
本文综述了计算环面簇 Chow 环的四种组合方法，并以此给出拟阵约化特征多项式系数公式的四个证明，揭示拟阵组合学的不同侧面，为拟阵交集理论提供发展框架。
Intersection theory of polymatroids
本文通过引入多拟阵的增广Chow环，推广了研究拟阵的代数几何方法，并证明其交数由Hall--Rado条件决定，该结果在拟阵情形下亦为新发现。
The rank-nullity ring of a matroid
本文提出 matroid 的秩-空环概念，并在均匀 matroid 情况下计算其 Hilbert 函数和 Gröbner 基。
Tropical Feichtner-Yuzvinsky and positivity criterion for fans
本文建立单纯扇形 Chow 环与其典范紧化热带上同调中间次数部分的同构，证明热带版 Kleiman 丰富性判据，并给出拟阵 Chow 环的新表示。
In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott-Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids.
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciproca…
Algebraic cobordism rings of wonderful varieties and matroids
本文给出了无环拟阵的Bergman扇的环面簇的代数配边环的两个组合表示，并证明了其与拟阵的Chow环和张量积的同构，推广了Chow环与K-环之间的积分同构。
The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an algorithm for computing the intersection degree of tuples of Keel's divisor classes. This computation is a concrete but complicated algorithmic question in the field. Also, we give a simple complexity argument for the algorithm. Additionally, we introduce three identities on multinomial coefficients, as well as proofs for them.
We study a particular graded ring structure on the set of all loopfree matroids on a fixed labeled ground set, which occurs naturally in tropical geometry. The product is given by matroid intersection and the additive structure is defined by assigning to each matroid the indicator vector of its chains of flats. We show that this ring is generated in corank one, more precisely that any matroid can be written as a linear combination of products of corank one matroids. Moreover, we prove that a basis for the graded part of rank r matroids is given by the set of nested matroids and that the total …
Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is …
The Chow group of zero cycles in the moduli space of stable pointed curves of genus zero is isomorphic to the integer additive group. Let $M$ be monomial in this Chow group. If no two factors of $M$ fulfill a particular quadratic relation, then the monomial can be represented equivalently by a specific tree; otherwise, $M$ is mapped to zero under the stated isomorphism. Starting from this tree representation, we introduce a graphical algorithm for computing the corresponding integer for $M$ under the aforementioned isomorphism. The algorithm is linear with respect to the size of the tree.
Chow rings of moduli spaces of genus 0 curves with collisions
本文通过新方法为0曲率模空间的Chow环提供了新的表示，并证明了Keel的结论。
The purpose of this paper is to prove that integral Chow rings of toric stacks are naturally isomorphic to Stanley-Reisner rings.
In this paper we compute the Chow ring of the moduli stack $\bar{M}_2$ of stable curves of genus 2 with integral coefficients.
In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.
In this paper we give a complete description of the integral Chow ring of the stack $\mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $\mathscr{H}_g$. We also give a geometric interpretation for the generators.
In this paper we prove that the rational Chow rings of $\mathcal{M}_{5,8}$ and $\mathcal{M}_{5,9}$ are tautological, and that these moduli spaces have the Chow Künneth generation Property.
We determine the Chow ring of the stack of truncated displays and more generally the Chow ring of the stack of G-zips. We also investigate the pull-back morphism of the truncated display functor. From this we can determine the Chow ring of the stack of truncated Barsotti-Tate groups over a field of characteristic p up to p-torsion.
Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring A\_T^*(H^d(X))\_Q and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
The Integral Chow Ring of Smooth Non-strict Toric Stacks
研究将Stanley-Reisner环扩展至非严格堆叠风扇，证明其与光滑非严格堆叠的积分Chow环同构，推广了更一般堆叠的Chow环结果。
We show how to translate the task of computing the multiplicative structure of a Chow ring of a projective homogeneous variety into an easily understandable combinatorial task of calculating in the corresponding polynomial ring. The algorithms are also presented as a Maple package. Then we proceed to compute the multiplicative structure of the Chow rings for projective homogeneous varieties E6/P1, E7/P7, and E8/P8.
We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.
We prove that Küchle fourfolds $X$ of type d3 have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of $X$.
In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.
Though the Chow group of 0-cycles on a K3 surface is quite large, we observe that the subgroup generated by product of divisors is cyclic.
We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.
We show this Chow ring is $\Z \oplus \Z$. We do this by partitioning the space into 2n subvarieties each of which is fibered over $Gl(2n-2,\C)/SO(2n-2,\C)$.
We prove that certain Fano fourfolds of K3 type constructed by Fatighenti-Mongardi have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.
In this paper we explicit the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.
The goal of this paper is to compute the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
Let $g$ be an even positive integer. In this paper we compute the integral Chow ring of the stack of smooth hyperelliptic curves of genus $g$.
We find a new presentation of the stack of hyperelliptic curves of odd genus as a quotient stack and we use it to compute its integral Chow ring by means of equivariant intersection theory.
We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.
Chow ring of moduli spaces of quasi-polarised K3 surfaces in lower genus
本文研究低亏格拟极化 K3 曲面模空间，证明亏格不超过 5 时其周环由重言类构成。
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.
For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and $\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings, extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen \cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$. We also describe an {\em asymptotic} product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle $V$ go to infinity…
We give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.
We show how the stratification method, introduced by Vezzosi in his study of PGL_3, provides a unified approach to the known computations of the Chow rings of the classifying spaces of GL_n, SL_n, Sp_n, O_n and SO_n.
We give a short proof of a presentation of the Chow ring of the Fulton-MacPherson compactification of n points on an algebraic variety. The result can be found already in Fulton and MacPherson's original paper. However, there is an error in one of the lemmas used in their proof. In the process we also determine the Chow rings of weighted Fulton--MacPherson compactifications.
We describe the Chow ring with rational coefficients of the moduli space of stable maps with marked points Mbar_{0,m}(n,d) as the subring of invariants of a ring B, relative to the action of the group of symmetries of d elements. B is computed by following a sequence of intermediate spaces for Mbar_{0,m}(n,d) and relating them to substrata of Mbar_{0,1}(n,d+m-1). An additive basis for the Chow ring is presented.
Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim in arXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of Chow motives and Chow rings of these configuration spaces.
We show that the Hopf algebra of quasi-symmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations of line bundles. In particular, we exhibit a gluing map which gives rise to the comultiplication. We then apply the result to calculate the Chow rings of certain stacks of semistable curves.
The Singular Cohomology Ring of a Matroid
本文引入了组合结构的单点上同调环，扩展了组合结构的Chow环。通过Adiprasito等人提出的组合翻转，证明了上同调环的消失定理，并计算了上同调的维数。对于均匀组合结构，给出了Hodge数的递归公式。最后，将单点上同调环推广到任意的格子平坦结构，并展示了上同调如何依赖于该结构。
In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.
In this paper we show that the family of stable vector bundles gives a set of generators for the Chow ring, the K-theory and the derived category of any smooth projective variety.
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