The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership problem for the symbolic powers $I_t^{(s)}$ and we compute the primary decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous Rees algebra \Rees(I_{t_1},\...,I_{t_k}) is defined by a Gr\"obner basis of quadrics

Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.Comment: 21 page

The standard graded property for vertex cover algebras of Quasi-Trees

J. Herzog, T. Hibi, N. V. Trung and X. Zheng characterize the vertex cover algebras which are standard graded. In this paper we give a simple combinatorial criterion for the standard graded property of vertex cover algebras in the case of quasi-trees. We also give an example of how this criterion works and compute the maximal degree of a minimal generator in that case