arXiv subjects

Phong Dinh Thieu

Publications and source records attributed to Phong Dinh Thieu.

3 recordsLinked to original sources

arXiv2014

Koszul determinantal rings and $2\times e$ matrices of linear forms

Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a $2\times e$ matrix of linear forms over a polynomial ring $k[\mathsf{x}_1, \ldots,\mathsf{x}_n]$ (where $e,n\ge 1$). We prove that the determinantal ring $R = k[\mathsf{x}_1,\ldots,\mathsf{x}_n]/I_2(X)$ is Koszul if and only if in the Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due to Conca.

math.AC↗

arXiv2013

On the weak Lefschetz Property of graded modules over $K[x,y]$

It is known that graded cyclic modules over $S=K[x,y]$ have the Weak Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The purpose of this note is to study which conditions on $S$-modules ensure the WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert function. In particular, we prove that indecomposable graded modules over $S$ with the Hilbert function $(h_0,h_1)$ have the WLP.

math.AC↗

arXiv2011

Note on resonance varieties

We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linear. Higher resonance varieties are also considered. We prove results supporting a conjecture of Schenck-Suciu relating the Betti numbers of the linear strand of J and its first resonance variety. A counter example is constructed that this conjecture is not true for arbitrary graded ideals.

math.AC↗