Mumford Regularity Research Articles

Castelnuovo–Mumford regularity and splitting criteria for logarithmic bundles over rational normal scroll surfaces

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for rational normal scroll surfaces. In this setting we prove analogs of some classical properties. We prove splitting criteria for coherent sheaves and a characterization of Ulrich bundles. Finally we study logarithmic bundles associated to arrangements of lines and rational curves.

Generalized binomial edge ideals of bipartite graphs

Connected bipartite graphs whose binomial edge ideals are Cohen–Macaulay have been classified by Bolognini et al. In this paper, we compute the depth, Castelnuovo–Mumford regularity, and dimension of the generalized binomial edge ideals of these graphs.

Castelnuovo–Mumford regularity of matrix Schubert varieties

Castelnuovo–Mumford regularity of matrix Schubert varieties

Continuous CM-regularity and generic vanishing

Abstract We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, O X (1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative for many pairs (X, O X (1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously k-regular for some positive integer k ≤ dim X, then F is a GV−(k−1) sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (X, O X (1)), and we show that this function can be extended to a continuous function on N 1(X)ℝ. We also provide syzygetic consequences of our results for Oℙ(E)(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties. In particular, we show that Oℙ(E)(1) satisfies the Np property if the base-point freeness threshold of the class of O X (1) in N 1(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito.

Cohomology of line bundles on the incidence correspondence

For a finite dimensional vector space V V of dimension n n , we consider the incidence correspondence (or partial flag variety) X ⊂ P V × P V ∨ X\subset \mathbb {P}V \times \mathbb {P}V^{\vee } , parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X X in characteristic p > 0 p>0 . If n = 3 n=3 then X X is the full flag variety of V V , and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0 0 , the cohomology groups are described for all V V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n = 3 n=3 , we recover the recursive description of characters from recent work of Linyuan Liu, while for general n n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

Determinantal Varieties From Point Configurations on Hypersurfaces

Abstract We consider the scheme $X_{r,d,n}$ parameterizing $n$ ordered points in projective space $\mathbb {P}^{r}$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure, and we prove that it is irreducible, Cohen–Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of $X_{r,d,n}$ in terms of Castelnuovo–Mumford regularity and $d$-normality. This yields a characterization of the singular locus of $X_{2,d,n}$ and $X_{3,2,n}$.

Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a_0,ldots ,a_{n-1}} be a finite set of nge 4 non-negative relatively prime integers, such that 0=a_0<a_1<cdots <a_{n-1}=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve mathcal {C}_A parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\quad \\mathcal {C}_A=\\{(v^d:u^{a_1}v^{d-a_1}:\\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \\mid (u:v)\\in \\mathbb {P}^{1}_k\\}\\subset \\mathbb {P}^{n-1}_k. \\end{aligned}$$\\end{document}The exponents in the previous parametrization of mathcal {C}_A define a homogeneous semigroup mathcal {S}subset mathbb {N}^2. We provide several results relating the Castelnuovo–Mumford regularity of mathcal {C}_A to the behavior of the sumsets of A and to the combinatorics of the semigroup mathcal {S} that reveal a new interplay between commutative algebra and additive number theory.

Weighted homological regularities

Let A A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A A -modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones.

Combinatorial Bounds of the Regularity of Elimination Ideals

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square–free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo–Mumford regularity of elimination ideals corresponding to certain family of graphs.

On the Castelnuovo–Mumford regularity of squarefree powers of edge ideals

Assume that G is a graph with edge ideal I(G) and matching number match(G). For every integer s≥1, we denote the s-th squarefree power of I(G) by I(G)[s]. It is shown that for every positive integer s≤match(G), the inequality reg(I(G)[s])≤match(G)+s holds provided that G belongs to either of the following classes: (i) very well-covered graphs, (ii) semi-Hamiltonian graphs, or (iii) sequentially Cohen-Macaulay graphs. Moreover, we prove that for every Cameron-Walker graph G and for every positive integer s≤match(G), we have reg(I(G)[s])=match(G)+s.

Ombinatorial Bounds of the Regularity of Elimination Ideals

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square--free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo--Mumford regularity of elimination ideals corresponding to certain family of graphs.

Regularity and symbolic defect of points on rational normal curves

In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective 3-space. We give an explicit formula for the value of Castelnuovo–Mumford regularity for their ordinary powers. Moreover, we compare the m-th symbolic and ordinary powers for such ideals in order to show whenever the m-th symbolic defect is non-zero.

GENERIC LINES IN PROJECTIVE SPACE AND THE KOSZUL PROPERTY

AbstractIn this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb {P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb {P}^n.$ We show that if $\mathcal {M}$ is a collection of m lines in general linear position in $\mathbb {P}^n$ with $2m \leq n+1$ and R is the coordinate ring of $\mathcal {M},$ then R is Koszul. Furthermore, if $\mathcal {M}$ is a generic collection of m lines in $\mathbb {P}^n$ and R is the coordinate ring of $\mathcal {M}$ with m even and $m +1\leq n$ or m is odd and $m +2\leq n,$ then R is Koszul. Lastly, we show that if $\mathcal {M}$ is a generic collection of m lines such that $$ \begin{align*} m&gt; \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$ then R is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n \leq 6$ or $m \leq 6$ . We also determine the Castelnuovo–Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.

Projective toric varieties of codimension 2 with maximal Castelnuovo–Mumford regularity

The Eisenbud–Goto conjecture states that reg X ≤ deg X − codim X + 1 for a nondegenerate irreducible projective variety X over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when X is a projective toric variety of codimension 2. We classify the projective toric varieties of codimension 2 having maximal regularity, that is, for which equality holds in the Eisenbud–Goto bound. We also give combinatorial characterizations of the arithmetically Cohen–Macaulay toric varieties of maximal regularity in characteristic 0.

Resolutions of ideals of subspace arrangements

Given a collection of t subspaces in an n-dimensional vector space W we can associate to them t linear ideals in the symmetric algebra 𝒮(W∗). Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of t linear ideals is at most t. We show that analogous results hold when we work over the exterior algebra ⋀(W∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.

Classes of cut ideals and their Betti numbers

We study monomial cut ideals associated to a graph G, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo–Mumford regularities are investigated if the graph can be decomposed as 0-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.

The v-number and Castelnuovo–Mumford regularity of graphs

We prove that for every integer \(k\ge 1\), there exists a connected graph \(H_k\) such that \(v(H_k)={\text {reg}}(H_k)+k\), where v(G) and \({\text {reg}}(G)\) denote the v-number and the (Castelnuovo–Mumford) regularity of a graph G respectively.

Eulerian ideals

Let G be a simple graph and , where is the homomorphism that sends an edge to the product of its vertices. The ideal is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Gröbner basis for , and we characterize the standard monomials of the ideal in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of , for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join.

Inequalities of invariants on Stanley–Reisner rings of Cohen–Macaulay simplicial complexes

The goal of the present paper is the study of some algebraic invariants of Stanley–Reisner rings of Cohen–Macaulay simplicial complexes of dimension d − 1 . We prove that the inequality d ≤ reg ( Δ ) ⋅ type ( Δ ) holds for any ( d − 1 ) -dimensional Cohen–Macaulay simplicial complex Δ satisfying Δ = core ( Δ ) , where reg ( Δ ) (resp. type ( Δ ) ) denotes the Castelnuovo–Mumford regularity (resp. Cohen–Macaulay type) of the Stanley–Reisner ring k [ Δ ] . Moreover, for any given integers d , r , t satisfying r , t ≥ 2 and r ≤ d ≤ r t , we construct a Cohen–Macaulay simplicial complex Δ ( G ) as an independent complex of a graph G such that dim ⁡ ( Δ ( G ) ) = d − 1 , reg ( Δ ( G ) ) = r and type ( Δ ( G ) ) = t .

Feasibility criteria for high-multiplicity partitioning problems

For fixed weights w 1 , ⋯ , w n , and for d > 0 , we let B denote a collection of d ⋅ n balls, with d balls of weight w i for each i = 1 , ⋯ , n . We consider the problem of assigning the balls to n bins with capacities C 1 , ⋯ , C n , in such a way that each bin is assigned d balls, without exceeding its capacity. When d ≫ 0 , we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w i are distinct. The feasibility criteria that we present here are used elsewhere (in commutative algebra) to study the asymptotic behavior of the Castelnuovo–Mumford regularity of symmetric monomial ideals.