The Castelnuovo-Mumford polynomials are the maximal degree components of Grothendieck polynomials. The support of each Castelnuovo-Mumford polynomial is conjectured to be M-convex, i.e. the set of integer points of a generalized permutahedron (Mészáros and St. Dizier, 2020). This conjecture is known to hold in certain special cases but remains open in general. We define new families of permutations whose Castelnuovo-Mumford polynomials we show to have M-convex support. Specifically, we investigate which permutations have Castelnuovo-Mumford polynomials whose supports are the set of integer points in a schubitope.
We prove that for every integer $k\geq 1$, there exists a connected graph $H_k$ such that $v(H_k)=reg(H_k)+k$, where $v(G)$ and $reg(G)$ denote the $v$-number and the (Castelnuovo-Mumford) regularity of a graph $G$ respectively.
We will give an upper bound for the $k$-normality of very ample lattice simplices, and then give an Eisenbud-Goto-type bound for some special classes of projective toric varieties.
We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.
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.
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo-Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo-Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo-Mumfor…
We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.
We study the Castelnuovo-Mumford regularity of tangent cones of Schubert varieties. Conjectures about this statistic are presented; these are proved for the covexillary case. This builds on work of L. Li and the author on these tangent cones, as well as that of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt and J. Rajchgot-C. Robichaux-A. Weigandt on tableau rules for computing regularity of some matrix Schubert varieties.
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Gr…
We prove that the projective dimension of any (hyper)graph can be bounded from above by the (Castelnuovo-Mumford) regularity of its Levi graph (or incidence bipartite graph). This in particular brings the use of regularity's upper bounds on the calculation of projective dimension of (hyper)graphs. When G is just a (simple) graph, we prove that there exists an induced subgraph H of G such that prod-dim(G)=reg(S(H)), where S(H) is the subdivision graph of H. Moreover, we show that known upper bounds on prod-dim(G) involving domination parameters are in fact upper bounds to reg(S(G)).
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 graph can be bounded from above by a function of its induced matching number. We also prove that the regularity of a graph is always less than or equal to the sum of its induced matching and decycling numbers.
In this paper we give bounds on the Castelnuovo-Mumford regularity of products of ideals and ideal sheaves. In particular, we show that the regularity of a product of ideals is bounded by the sum of the regularities of its factors if the corresponding schemes intersect in a finite set of points. We also show how approximations of sheaves can be used to bound the regularity of an arrangement of two-planes in projective space.
We present a simple method to obtain information regarding the connectivity of the 1-skeleta of a wide family of simplicial complexes through bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. In this way we generalize and unify two results on connectivity: one by Balinsky and Barnette, one by Athanasiadis. In particular, if $Δ$ is a simplicial $d$-pseudomanifold, and $s$ is the highest integer such that there is an $s$-dimensional simplex not contained in $Δ$, but such that its boundary is, then the 1-skeleton of $Δ$ is $\left\lceil \frac{(s+1)d}{s} \right\rceil$-co…
We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…
The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.
We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, cho…
Assume that $G$ is a graph with edge ideal $I(G)$ and matching number ${\rm match}(G)$. For every integer $s\geq 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G)^{[s]}$. It is shown that for every positive integer $s\leq {\rm match}(G)$, the inequality ${\rm reg}(I(G)^{[s]})\leq {\rm match}(G)+s$ 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 $s\leq {\rm match}(G)$, we …
Castelnuovo-Mumford regularity of toric varieties with at most one singular point
Universidad de La Laguna（拉古纳大学）；Instituto de Matemáticas y Aplicaciones (IMAULL), Sección de Matemáticas, Facultad de Ciencias, Universidad de La Laguna（拉古纳大学理学院数学系数学与应用数学研究所 (IMAULL)）；Universidad de Valladolid（巴利亚多利德大学）；Instituto de Investigación en Matemáticas de la Universidad de Valladolid (IMUVA), Universidad de Valladolid（巴利亚多利德大学）
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, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with …
We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute the Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs.
We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases.
Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove, improve and generalize many of the known bounds on regularity of quadratic square-free monomial ideals.
We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a key role in proof of the base case s =2.
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$
We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.
We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.
