Showing papers on "Castelnuovo–Mumford regularity published in 2018"
TL;DR: In this article, it was shown that if G is a gap-free and diamond-free graph, then I(G)s has a linear minimal free resolution for every s ≥ 2.
Abstract: We show that if G is a gap-free and diamond-free graph, then I(G)s has a linear minimal free resolution for every s≥2.
20 citations
TL;DR: In this article, the existence of F-thresholds in full generality was shown in a generalization of standard graded algebras over a field for which F-pure threshold and F -threshold at the irrelevant maxi...
Abstract: We show the existence of F-thresholds in full generality. In addition, we study properties of standard graded algebras over a field for which F-pure threshold and F-threshold at the irrelevant maxi ...
19 citations
TL;DR: In this paper, it was shown that the -invariant, the -pure threshold, and the diagonal threshold hold only assuming that the algebra is -pure, and they were conjectured for strongly regular rings.
Abstract: The -invariant, the -pure threshold, and the diagonal -threshold are three important invariants of a graded -algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly -regular rings. In this article, we prove that these relations hold only assuming that the algebra is -pure. In addition, we present an interpretation of the -invariant for -pure Gorenstein graded -algebras in terms of regular sequences that preserve -purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition . We also present analogous results and questions in characteristic zero.
17 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups.
Abstract: We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.
15 citations
TL;DR: In this article, it was shown that for arbitrary integers r and s with r ≥ 1 and s ≥ 1, a monomial ideal I of S = K[x1,…,xn] with n ≥ 0 for which reg(S/I)=r and degh S/I(λ)=s will be constructed.
Abstract: Let S=K[x1,…,xn] denote the polynomial ring in n variables over a field K with each degxi=1 and let I⊂S be a homogeneous ideal of S with dimS/I=d. The Hilbert series of S/I is of the form hS/I(λ)/(1−λ)d, where hS/I(λ)=h0+h1λ+h2λ2+⋯+hsλs with hs≠0 is the h‐polynomial of S/I. It is known that, when S/I is Cohen–Macaulay, one has reg(S/I)=deghS/I(λ), where reg(S/I) is the (Castelnuovo–Mumford) regularity of S/I. In the present paper, given arbitrary integers r and s with r≥1 and s≥1, a monomial ideal I of S=K[x1,…,xn] with n≫0 for which reg(S/I)=r and deghS/I(λ)=s will be constructed. Furthermore, we give a class of edge ideals I⊂S of Cameron–Walker graphs with reg(S/I)=deghS/I(λ) for which S/I is not Cohen–Macaulay.
14 citations
TL;DR: New combinatorial insights are presented into the calculation of (Castelnuovo-Mumford) regularity of graphs.
Abstract: We present new combinatorial results on the calculation of (Castelnuovo-Mumford) regularity of graphs. We introduce the notion of a prime graph over a field k, which we define to be a connected graph with regk(G − x) < regk(G) for any vertex x ∈ V (G). We then exhibit some structural properties of prime graphs. This enables us to provide upper bounds to the regularity involving the induced matching number im(G). We prove that reg(G) ≤ (Γ(G)+1)im(G) holds for any graph G, where Γ(G)=max{|N G [x]\N G [y]| : xy ∈ E(G)} is the maximum privacy degree of G and N G [x] is the closed neighbourhood of x in G. In the case of claw-free graphs, we verify that this bound can be strengthened by showing that reg(G)≤2im(G). By analysing the effect of Lozin transformations on graphs, we narrow the search for prime graphs into graphs having maximum degree at most three. We show that the regularity of such graphs G is bounded above by 2im(G)+1. Moreover, we prove that any non-trivial Lozin operation preserves the primeness of a graph. That enables us to generate many new prime graphs from the existing ones.
8 citations
TL;DR: In this paper, the authors established strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal and proved that the two regularities are equal for large classes of monomial ideals in two variables.
7 citations
Posted Content•
TL;DR: In this article, upper and lower bounds on the Castelnuovo-Mumford regularity are given in terms of the Hilbert coefficients of the coefficients of a Hilbert function.
Abstract: New upper and lower bounds on the Castelnuovo-Mumford regularity are given in terms of the Hilbert coefficients Examples are provided to show that these bounds are in some sense nearly sharp
2 citations
TL;DR: In this paper, an upper bound for Castelnuovo-Mumford regularity of I by the big-size of I was given, where the projective dimension of I is bounded by the smallest number t with the property that for all integers 1 6 i1 < i2 <... < i 6 m such that
Abstract: Let S = K[x1; x2;...; xn] be the polynomial ring in n variables over a field K; and let I be a squarefree monomial ideal minimally generated by the monomials u1; u2;...; um: Let w be the smallest number t with the property that for all integers 1 6 i1 < i2 <... < i
t
6 m such that $$lcm({u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_t}}}) = lcm({u_1},{u_2},...,{u_m})$$
We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I: As a corollary, the projective dimension of I is bounded by the number w.
1 citations
Dissertation•
02 Jul 2018
TL;DR: In this article, the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes was studied, and it was shown that the Castelnuovo-Mumford regularity of normal projective toric surfaces is k-normal.
Abstract: We study the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes. In particular, we give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties. We also give a new combinatorial proof for a special case of Reider’s Theorem for smooth toric surfaces.
1 citations
TL;DR: In this article, the regularity and projective dimension of edge ideals of vertex decomposable graphs were studied and two upper bounds for the degree of regularity of the edge ideals were derived.
Abstract: In this paper, we study the regularity and projective dimension of edge ideals. We provide two upper bounds for the regularity of edge ideals of vertex decomposable graphs in terms of the i...
16 Feb 2018
TL;DR: In this article, a relation between the regularity of Koszul cycles and homologies of a homogeneous ideal in a polynomial ring was established, and a sharp regularity bound for the homology of the Koszula homology was shown.
Abstract: We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.