arXiv 1102.25472011-02math.ACmath.AGmath.CO

The geometry and combinatorics of cographic toric face rings

Sebastian Casalaina-Martin, Jesse Leo Kass, Filippo Viviani

备注：: 27 pages; final version, to appear in Algebra & Number Theory
发表信息：: Algebra Number Theory 7 (2013), no. 8, 1781--1815
DOI：: 10.2140/ant.2013.7.1781

英文摘要：

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed from graphs. Among the rings that can be constructed from a graph, cographic rings are particularly interesting because they appear in the study of compactified Jacobians of nodal curves.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study the combinatorics of Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterizatio…

2020-09-12math.ACmath.CO

Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomolo…

2019-12-29math.ACmath.ATmath.CO

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for $F$-pure rings that were inspired by work done for Stanley-Reisner rings.

2022-03-18math.AC

Segre embeddings, Hilbert series and Newcomb's problem

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original questio…

2014-01-16math.ACmath.AG

Toric Topology

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discov…

2014-07-04math.ACmath.AGmath.AT

Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

There are two seemingly unrelated ideals associated with a simplicial complex Δ. One is the Stanley-Reisner ideal I_Δ, the monomial ideal generated by minimal non-faces of Δ, well-known in combinatorial commutative algebra. The other is the toric ideal I_{M(Δ)} of the facet subring of Δ, whose generators give a Markov basis for the hierarchical model defined by Δ, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I_{M(Δ)} is determined by the Betti numbers of I_Δ. The unexpected connection between the syzygies of the Stanley-Reisner…

2012-05-31math.ACmath.COmath.ST

更多相关论文收起更多论文

关于环面流形的上同调

On the cohomology of torus manifolds

环面流形是由具有非空不动点集和一些附加方向数据的半维环面作用的偶数维流形。它可以被认为是一个深远的概括环面流形从代数几何。环面流形的轨道空间具有丰富的组合结构，例如，它是一个带有角的歧管，只要动作是当地标准的。在这里，我们研究环面流形的上同调性质和它们的轨道向量的组合之间的关系。证明了环面流形的上同调环是由二维类生成的当且仅当其商是同调多胞形。在这种情况下，我们从复曲面几何中恢复熟悉的图像：等变上同调是神经单纯复形的面环，而普通上同调是通过分解出某些线性型而得到的。在更一般的情况下，我们证明了环面流形的奇次上同调为零当且仅当其轨道空间是面非循环的。虽然在这种情况下上同调不再是二次生成的，但等变上同调仍然同构于适当单纯偏序集的面环。

2006-10-12math.ACmath.ATmath.CO

Elementary symmetric polynomials in Stanley--Reisner face ring

Let $P$ be a simple polytope of dimension $n$ with $m$ facets. In this paper we pay our attention on those elementary symmetric polynomials in the Stanley--Reisner face ring of $P$ and study how the decomposability of the $n$-th elementary symmetric polynomial influences on the combinatorics of $P$ and the topology and geometry of toric spaces over $P$. We give algebraic criterions of detecting the decomposability of $P$ and determining when $P$ is $n$-colorable in terms of the $n$-th elementary symmetric polynomial. In addition, we define the Stanley--Reisner {\em exterior} face ring $\mathca…

2016-02-29math.ACmath.ATmath.CO