Related papers: A bound for Castelnuovo-Mumford regularity by doub…

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gr\"obner basis of $I$ with respect to the degree…

We study the homological algebra of edge ideals of Erd\"{o}s-R\'enyi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some…

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where…

Weighted Castelnuovo-Mumford Regularity and Weighted Global GenerationWe introduce and study a notion of Castelnuovo-Mumford regularity suitable for weighted projective spaces.

We prove that when a Lozin's transformation is applied to a graph, the (Castelnuovo-Mumford) regularity of the graph increases exactly by one, as it happens to its induced matching number. As a consequence, we show that the regularity of a…

We discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of linear…

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the length of an $R_\eta$-sequence containing fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result…

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$,…

We prove a sharp inequality relating the Castelnuovo--Mumford regularity of a coherent ideal sheaf to its log-canonical threshold.

We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not to be Noetherian.

We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed…

Let $G$ be a simple graph and $I(X_G)=\varphi^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $\varphi \colon K[E_G]\to K[V_G]$ is the homomorphism that sends an edge to the product of its vertices. The ideal $I(X_G)$ is Cohen--Macaulay,…

We establish bounds for the Castelnuovo-Mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations. We extend to modules (and…

Let n,d be positive integers, with d even (say d=2e). Let X_(n,d) denote the locus of degree d hypersurfaces in P^n which consist of two e-fold hyperplanes. We bound the regularity of the ideal of this variety. Moreover, we show that this…

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free…

Let $X^n \subset P^N$ be a nonsingular, nondegenerate projective variety of dimension $n$ and codimension $N-n \ge 2$. Let $|C_X|$ be the linear system determined by the double-point divisor obtained by generically projecting $X$ to a…

For all integers $4 \leq r \leq d$, we show that there exists a finite simple graph $G= G_{r,d}$ with toric ideal $I_G \subset R$ such that $R/I_G$ has (Castelnuovo-Mumford) regularity $r$ and $h$-polynomial of degree $d$. To achieve this…

The purpose of this paper is to give a simple geometric construction of ideals whose Castelnuovo-Mumford regularity is large compared to the generating degree. Moreover, our ideals have the property that the Castelnuovo-Mumford regularity…

For a partition $\lambda$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. In the previous paper, the second author showed that if $R/I^{\rm…

The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction,…