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Let $\mathbf{a} = a_1 <\dots < a_r$ be a sequence of positive integers, and let $H_{\mathbf{a}}$ denote the semigroup generated by $a_1, \dots, a_r$. For an integer $k\geq 0$ we denote by $\mathbf{a}+k$ the shifted sequence $a_1 +k, \dots,…

Suppose X is any finite complex with vanishing L^2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and…

We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.

We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that…

We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb K}[x_1, \dots x_n]$, utilizing methods from the Erd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$ we consider…

We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$,…

For a standard graded Cohen-Macaulay ring $S$, if the quotient $S/(\underline{x})$ admits non-free totally reflexive modules, where $\underline{x}$ is a system of parameters consisting of elements of degree one, then so does the ring $S$.…

Let S=K[x_1,...,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I is a graded ideal in S. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the…

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…

The classical results, initiated by Castelnuovo and Fano and later refined by Eisenbud and Harris, provide several upper bounds on the number of quadrics defining a nondegenerate projective variety. Recently, it has been revealed that these…

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this…

Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s \geq 1. These…

Let Z be a finite set of double points in P^1 x P^1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X, for the bigraded…

Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for every i between 0 and p, the dimension of the K-vector space Tor^S_i(M,N) is at…

Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal…

Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa…

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at…

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be…

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary…

Let $k$ be a field. We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$. Also we compute the graded local cohomologies of the Rees rings $A[I t]$ and give the…