Showing papers on "Castelnuovo–Mumford regularity published in 2015"

The regularity of powers of edge ideals

Abstract: In this paper, we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order, we find new upper bounds on the regularity of powers of edge ideals of graphs whose complement does not have any induced four cycles.

The Castelnuovo-Mumford regularity of binomial edge ideals

Abstract: We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that $\mathrm{reg}(J_G)\leq n-1$ for any graph $G$ with $n$ vertices, which is not a path. Moreover, we study the behavior of the regularity of binomial edge ideals under the join product of graphs.

The Locus of Points of the Hilbert Scheme with Bounded Regularity

Abstract: In this paper we consider the Hilbert scheme parameterizing subschemes of ℙ n with Hilbert polynomial p(t), and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer r′. This locus is an open subscheme of and, for every s ≥ r′, we describe it as a locally closed subscheme of the Grasmannian given by a set of equations of degree ≤deg(p(t)) +2 and linear inequalities in the coordinates of the Plucker embedding.

Castelnuovo–Mumford regularity bounds for singular surfaces

Abstract: We prove the regularity conjecture, namely Eisenbud–Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree.

Castelnuovo-Mumford regularity of graphs

Abstract: We present new combinatorial insights into the calculation of (Castelnuovo-Mumford) regularity of graphs.

Minimal Castelnuovo–Mumford Regularity for a Given Hilbert Polynomial

Abstract: Let K be an algebraically closed field of null characteristic and p(z) a Hilbert polynomial. We look for the minimal Castelnuovo–Mumford regularity mp(z) of closed subschemes of projective spaces over K with Hilbert polynomial p(z). Experimental evidences led us to consider the idea that mp(z) could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo–Mumford regularity mϱp(z) of schemes with Hilbert polynomial p(z) and given regularity ϱ of the Hilbert function, and also the minimal Castelnuovo–Mumford regularity mu of schemes with Hilbert function u. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.

Segre Product, H-Polynomials, and Castelnuovo-Mumford Regularity

Abstract: The purpose of this paper is to extend the bilinearity of the Segre product that has been proved recently by Ilse Fischer and Martina Kubitzke under some restricted hypotheses. As a consequence, we get some formulas involving Eulerian polynomials and a nice formula that will be used by the first author to solve the Simon Newcomb problem. We apply these results to compute the postulation number of a series and extend partially the results about Castelnuovo-Mumford regularity of the Segre product of polynomial rings of David A. Cox and Evgeny Materov.

Castelnuovo-Mumford regularity and Ratliff-Rush closure

Abstract: We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases.

Cover product and betti polynomial of graphs

Abstract: The cover product of disjoint graphs and with fixed vertex covers and , is the graph with vertex set and edge set We describe the graded Betti numbers of in terms of those of and . As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph such that , where, denotes the edge ideal of , is the Castelnuovo–Mumford regularity of and is the induced or strong matching number of ; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The -vector of is described in terms of the -vectors of and . Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

Castelnuovo-Mumford Regularity and Edge Ideals

Abstract: In this dissertation we study the homological algebra of the monomial ideals with a special emphasis on the topics of the Castenuovo-Mumford regularity and the powers of edge ideals of finite simple graphs. The main problem of this dissertation is to find optimal bounds for the regularity of powers of edge ideals. To do this, we prove the existence of a very special order of the minimal monomial generators of powers of the edge ideal. Using this order and some short exact sequence techniques we prove that the regularity of a power of an edge ideal can be bounded by the maximum of the regularities of the edge ideals of some very closely related graphs, and as corollaries we show that for various classes of graphs the higher powers of edge ideals have linear minimal free resolutions. One of these corollaries partially answers a case of a conjecture proposed by Eran Nevo and Irena Peeva. In the process of this study we introduce a new notion called even connectedness in finite simple graphs and derive various results related to it. In particular, we show that this behaves particularly nicely in the case of bipartite graphs and prove some results related to regularity of powers of edge ideals of bipartite graphs. We also study path ideals of finite simple graphs in the same spirit and show that various classes of path ideals also have linear minimal free resolution. Using similar techniques we also study the Cohen-Macaulayness of bipartite edge ideals and prove a new characterization for it.

Castelnuovo-Mumford regularity of products of monomial ideals

Abstract: Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l}^{a_l})$.

Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings

Abstract: In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer l with 0≤l≤n−d−1. If e(K[Δ])≤(l+1)d−l and β l,l+d (K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

Castelnuovo-Mumford regularity of fiber cones of filtered modules

Abstract: We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cones which depend on the length of certain local cohomology modules. The bounds are the analogue of the ones proved by Dung and Hoa for the associated graded module of a filtered module. 2010 Mathematics Subject Classification: 13A30; 13D40