Lexsegment ideals of Hilbert depth 1

Let I\subset S=\KK[x_1,...,x_n] be a lexsegment ideal, generated by monomials of degree $d$. The main aim of this paper is to characterize when the Hilbert depth of $I$ will be 1, in the standard graded case. In addition to this, we will give an estimate of depth of squarefree monomial ideals, generalizing a result of Popescu. We will also show that Stanley conjecture holds for squarefree stable ideals, in the multigraded case.Comment: 14 page

Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

We investigate the Castelnuovo--Mumford regularity and the multiplicity of the toric ring associated to a three-dimensional Ferrers diagram. In particular, in the rectangular case, we are able to provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanied pair of Ferrers diagrams under some mild conditions, and bound the Castelnuovo--Mumford regularity for more general cases.Comment: 22 pages, 2 figures and comments are welcom