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.
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.
Weighted Castelnuovo-Mumford Regularity and Weighted Global GenerationWe introduce and study a notion of Castelnuovo-Mumford regularity suitable for weighted projective spaces.
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.
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 is revealed as late in the resolution as desired.
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.
The behaviour of Castelnuovo-Mumford regularity under ``geometric'' transformations is not well understood. In this paper we are concerned with examples which will shed some light on certain questions concerning this behaviour.
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.
Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are given. We also derive a bound on the homological degree in terms of the Castelnuovo-Mumford regularity. This answers positively a question raised by Vasconcelos.
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in P^n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.
In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
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 conjectu…
The main result of this paper shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.
Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo-Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo-Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.
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 improve) the known bounds for homogenous ideal in a polynomial ring established by Galligo, Guisti, Caviglia and Sbarra.
We define the concept of regularity for bigraded modules and bigraded polynomial ring. In this setting we prove analogs of some of the classical results on $m$-regularity for graded modules over polynomial algebras.
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.
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）
Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates, and holds for any field.
In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proof is to show that generic residual intersections of complete intersection rational singularities again have rational singularities. When applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals.
We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, under appropriate geometric hypotheses. Results on the rigidity of multiple Tor modules and on the characterization of their vanishing are given, motivated by geometric applications.
We give examples of nonsingular curves in projective 3 space such that the regularity of powers of their ideal sheaves are highly nonlinear. This is in constrast to the case of an ideal I in a polynomial ring, where the regularity of I^n is a linear polynomial in n for large n.
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.
We give a bound on the Castelnuovo-Mumford regularity of a homogeneous ideals I, in a polynomial ring A, in terms of number of variables and the degrees of generators, when the dimension of A/I is at most two. This bound improves the one obtained by Caviglia and Sbarra in this case. In the continuation of the examples constructed by Clare D'Cruz and the first author, we use families of monomial curves to construct homogeneous ideals showing that these bounds are quite sharp.
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 by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillma…
Bounds on the Castelnuovo-Mumford regularity of the associated graded modules of k-Buchsbaum modules M are given in terms of k and some other invariants of M.
We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not to be Noetherian.
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.
In this paper we give a nice formula for the Castelnuovo-Mumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).
Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic ini…
In this paper we show how, given a complex of graded modules and knowing some partial Castelnuovo-Mumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if $\dim \tor_1^R(M,N)\leq1$ then $\reg(M\otimes N)\leq \reg(M)+\reg(N)$, generalizing results of Chandler, Conca and Herzog, and Sidman. Finally we give a description of the regularity of a module in terms of the postulation numbers of filter regular hyperplane restrictions.
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 show that bounds on the Castelnuovo-Mumford regularity of singular schemes, as a function of the degrees of the equations defining the shceme, of its dimension and of the dimension of their singular space. In the case where the singularities are isolated, we improve the bound given by Chardin and Ulrich, and in the general case we establish a bound doubly exponential in the dimension of the singular space. -- Nous montrons dans cet article des bornes pour la regularite de Castelnuovo-Mumford d'un schema admettant des singularites, en fonction des degres des equations definissant le schema, …
An upper bound for the Castelnuovo-Mumford regularity of the associated graded module of an one-dimension module is given in term of its Hilbert coeffcients. It is also investigated when the bound is attained.
We give bounds on the Castelnuovo-Mumford regularity of the associated graded module of an arbitrary good filtration and of its fiber cone. These bounds extend previous results of Rossi-Trung-Valla and Linh.
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 show that the Eisenbud-Goto conjecture holds for (homogeneous) seminormal simplicial affine semigroup rings. Moreover, we prove an upper bound for the Castelnuovo-Mumford regularity in terms of the dimension, which is similar as in the normal case. Finally, we compute explicitly the regularity of full Veronese rings.
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 prove the Eisenbud-Goto conjecture for connected curves. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in projective 4-space having maximal regularity, but no extremal secants. We also show that any connected curve in projective 3-space of degree at least 5 that has no linear components and has maximal regularity has an extremal secant.
We give two kinds of bounds for the Castelnuovo-Mumford regularity of the canonical module and the deficiency modules of a ring, respectively in terms of the homological degree and the Castelnuovo-Mumford regularity of the original ring.
In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exists for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $2\times 2$ minors in a $2\times 3$ matrix of indeterminates form d…
In this article, we study the Castelnuovo-Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.
We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated $\mm$-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud and Ulrich which states that these regularities are equal.
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…
These notes are an introduction to some basic aspects of the Castelnuovo-Mumford regularity and related topics such as weak regularity, a*-invariant and partial regularities.
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the assoc…
In this paper we show that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring for the good filtrations case. We obtain relationships between the Castelnuovo-Mumford regularity of the fiber cone, associated graded ring, Rees algebra and reduction number for the good filtrations case.
We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension.
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…
In a recent preprint, Ilse Fischer and Martina Kubitzke, proved the bilinearity of the Segre transform under some restricted hypothesis, motivated by their results we show in this paper the bilinearity of the Segre transform in general. We apply these results to compute the postulation number of a series. Our second application is motivated by the paper of David A. Cox, and Evgeny Materov (2009), where is computed the Castelnuovo-Mumford regularity of the Segre Veronese embedding, we can extend partially their result and compute the Castelnuovo-Mumford regularity of the Segre product of Cohen-…
