Castelnuovo Mumford Regularity Research Articles

Regularity of the edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

Solving Degree Bounds for Iterated Polynomial Systems

For Arithmetization-Oriented ciphers and hash functions Gröbner basis attacks are generally considered as the most competitive attack vector. Unfortunately, the complexity of Gröbner basis algorithms is only understood for special cases, and it is needless to say that these cases do not apply to most cryptographic polynomial systems. Therefore, cryptographers have to resort to experiments, extrapolations and hypotheses to assess the security of their designs. One established measure to quantify the complexity of linear algebra-based Gröbner basis algorithms is the so-called solving degree. Caminata & Gorla revealed that under a certain genericity condition on a polynomial system the solving degree is always upper bounded by the Castelnuovo-Mumford regularity and henceforth by the Macaulay bound, which only takes the degrees and number of variables of the input polynomials into account. In this paper we extend their framework to iterated polynomial systems, the standard polynomial model for symmetric ciphers and hash functions. In particular, we prove solving degree bounds for various attacks on MiMC, Feistel-MiMC, Feistel-MiMC-Hash, Hades and GMiMC. Our bounds fall in line with the hypothesized complexity of Gröbner basis attacks on these designs, and to the best of our knowledge this is the first time that a mathematical proof for these complexities is provided. Moreover, by studying polynomials with degree falls we can prove lower bounds on the Castelnuovo-Mumford regularity for attacks on MiMC, Feistel-MiMC and Feistel-MiMCHash provided that only a few solutions of the corresponding iterated polynomial system originate from the base field. Hence, regularity-based solving degree estimations can never surpass a certain threshold, a desirable property for cryptographic polynomial systems.

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.

Correction to: The v-Number and Castelnuovo-Mumford Regularity of Cover Ideals of Graphs

Correction to: The v-Number and Castelnuovo-Mumford Regularity of Cover Ideals of Graphs

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.

The v-Number and Castelnuovo-Mumford Regularity of Cover Ideals of Graphs

Abstract The $\textrm{v}$-number of a graded ideal $I\subsetneq R$, denoted by $\textrm{v}(I)$, is the minimum degree of a polynomial $f$ for which $I:f$ is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the $\textrm{v}$-number of edge ideals. In this paper, we study the $\textrm{v}$ number of the cover ideal $J(G)$ of a graph $G$. The main result shows that $\textrm{v}(J(G))\leq \textrm{reg}(R/J(G))$ for any simple graph $G$, which is quite surprising because, for the case of edge ideals, this inequality does not hold. Our main result relates $\textrm{v}(J(G))$ with the Cohen-Macaulay property of $R/I(G)$, where $I(G)$ denotes the edge ideal of $G$. We provide an infinite class of connected graphs, which satisfy $\textrm{v}(J(G))=\textrm{reg}(R/J(G))$. Also, we show that for every positive integer $k$, there exists a connected graph $G_{k}$ such that $\textrm{reg}(R/J(G_{k}))-\textrm{v}(J(G_{k}))=k$. Also, we explicitly compute the $\textrm{v}$-number of cover ideals of cycles.

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.

Duality for asymptotic invariants of graded families

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants.We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals.In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

Upper bounds on two Hilbert coefficients

New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of Hilbert series as well as in terms of the Castelnuovo-Mumford regularity of the associated graded module.

Regularity of primes associated with polynomial parametrisations

We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation.

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.

Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes

These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with [51]. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian schemes.

On vertex decomposability and regularity of graphs

There are two motivating questions in [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, arXiv:1006.1087v1] and [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, J. Pure Appl. Algebra, 215(10) (2011), 2473-2480] about Castelnuovo-Mumford regularity and vertex decomposability of simple graphs. In this paper, we give negative answers to the questions by providing two counterexamples.

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.

A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

We show that a bound of Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety X ⊆ P r is sharp exactly for complete intersections, provided the variety X is cut out scheme-theoretically by several hypersurfaces in P r . This generalizes a result of Bertram-Ein-Lazarsfeld.