A conjecture on the Poincar{é}-Betti series of modules of differential operators on a generic hyperplane arrangement

超平面构形雅可比行列式的几何性质

Geometric aspects of the Jacobian of a hyperplane arrangement

该研究发展超平面构形的弱透视表示理论，形式化并推广超平面构形与刚性理论间的关联，探究组合与几何的相互作用对构形相关代数结构的影响，建立两类研究间的联系。

Syzygies of Oriented Matroids

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…

关于一般纤维积环、Poincaré级数及其结构

On General fiber product rings, Poincaré series and their structure

本文研究一般纤维积环的结构，证明其上任意模的Poincaré级数受有理函数约束，刻画了环深度这一公开问题，并证明了其非正则性及对若干猜想的肯定回答。

接近自由构形的对数导子模的自由分解

Free resolution of the logarithmic derivation modules of close to free arrangements

该文研究从自由构形删除两个超平面得到的接近自由构形的对数导子模结构，给出其极小自由分解信息与分次Betti数下界，确定三维情形的极小自由分解并辅以示例揭示代数与组合性质的关联。

Schemes supported on the singular locus of a hyperplane arrangement in $\mathbb P^n$

We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's conjecture says that freeness of $\mathcal A$ is determined by the combinatorics of the intersection lattice of $\mathcal A$. We study the Cohen-Macaulayness of three other ideals, all unmixed, that are closely related to $\mathcal A$. Let $\overline J = \mathfrak q_1 \cap \dots \cap \mathfrak q_s$ be the intersection of height two primary com…

Hilbert series, Poincaré series and homotopy Lie algebras of graded algebras -- a seminar

We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincaré series for $A$ in two variables. This gives the Fröberg formula in the case where the bigraded $Tor^A(k,k)$ is concentrated on the diagonal, which we take as definition of $A$ being "Koszul". We look at a resolution in the commutative case obtained from the Koszul complex in the "trivially Golod" case. The algebra structure of $Ext_A(k,k)$ is introduced in different ways. Its subalgebra generated by the one-dimensional elements is by definition the "Koszul" dual of $A$. We define the"generalize…

Local Cohomology, Arrangement of Subspaces and Monomial Ideals

Let $k$ be a field of characteristic zero and I an ideal defining an arrangement of linear subspaces in the affine space $A^n_k$. We compute the D-module theoretic characteristic cycle of the local cohomology modules $H^r_I(k[x_1,...,x_n])$ in terms of the poset defined by the arrangement. In case I is a monomial ideal, we relate the multiplicities of the characteristic cycle with the Betti numbers of the Alexander dual ideal of I, we also study some extension problems attached to the modules $H^r_I(k[x_1,...,x_n])$. If $k$ is the field of complex numbers we study the ubiquity of these local c…