Related papers: On the facet ideal of an expanded simplicial compl…

Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $\lambda\in[0, 1]$, that the set of complex numbers…

We call the $\delta$-vector of an integral convex polytope of dimension $d$ flat if the $\delta$-vector is of the form $(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)$, where $a \geq 1$. In this paper, we give the complete characterization of…

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $\Delta$. We consider the more general notion of an $r$-DFI,…

Given an effective Q-divisor D on a smooth complex variety, one can associate to D its multiplier ideal sheaf J(D), which measures in a somewhat subtle way the singularities of D. Because of their strong vanishing properties, these ideals…

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$.…

We consider the face ideal associated to a line-type simplicial complex. We compute the \texttt{depth} and the \texttt{sdepth} for its quotient ring. In particular, the facet ideal and its quotient ring satisfy the Stanley inequality.

This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes…

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it…

A normal (respectively, graded normal) vector configuration $A$ defines the toric ideal $I_A$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when $A$ is normal and graded, $I_A$ is…

One of the most common and effective methods of obtaining structural information on simplicial complexes is to use tools from algebraic geometry/commutative algebra (often motivated by properties of toric varieties). However, there is no…

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture…

We prove a strengthening of the trickle down theorem for partite complexes. Given a $(d+1)$-partite $d$-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are $\frac{1-\delta}{d}$-(one-sided)…

In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge…

We give two formulae which express the Alexander polynomial $\Delta^C$ of several variables of a plane curve singularity $C$ in terms of the ring ${\cal O}_{C}$ of germs of analytic functions on the curve. One of them expresses $\Delta^C$…

We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid.…

We investigate the algebra and geometry of the independence conditions on discrete random variables in which we fix some random variables and study the complete independence of some subcollections. We interpret such independence conditions…

We study the regularized determinants ${\rm det}\, \Delta$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|\omega|^2$, where $\omega$ is a…

We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain K\"uhnel-type bounds on the Betti numbers as well as on certain…

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field. Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$, satisfies the $G_d$ condition, the weak Artin-Nagata property $AN_{d-2}^-$…