arXiv subjects

Giulio Caviglia

Publications and source records attributed to Giulio Caviglia.

At least 19 recordsLinked to original sources

arXiv2026

Explicit Stillman bounds for all degrees

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the length of an $R_η$-sequence containing fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result yields many other uniform bounds including bounds on the projective dimension of the ideals generated by $n$ forms of degree at most $d$. Explicit values of these bounds for forms of degree $5$ and higher are not yet known. This article constructs such explicit bounds, one of which is an upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by $n$ forms of degree at most $d$. In the settings of the Eisenbud-Goto conjecture, we derive an explicit bound of the Castelnuovo-Mumford regularity of a nondegenerate prime ideal $P$ in a polynomial ring $S$ in terms of the multiplicity of $S/P$.

math.AC↗

arXiv2026

Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies.

math.AC↗

arXiv2026

Quadratic linear strands of prime ideals

We prove sharp estimates on the quadratic strand of the resolution of any homogeneous prime ideal in a standard graded polynomial ring over an arbitrary field. Our bounds only depend on the height of the prime ideal, and they are optimal since for every $h \geq 1$ we show that there exists a prime ideal of height $h$ achieving them. In particular, we show that a prime ideal of height $h$ can contain at most $h^2$ quadratic minimal generators, and that there exists a prime ideal minimally generated by $h^2$ quadrics.

math.AC↗

arXiv2024

Bounds for syzygies of monomial curves

Let G be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of G which depends only on the width of G, that is, the difference between the largest and the smallest generator of G. In this way, we make progress towards a conjecture of Herzog and Stamate. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.

math.AC↗

arXiv2023

A criterion for sequential Cohen-Macaulayness

The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm gin}_{revlex}(U))$, where $F$ is a graded free $S$-module and $M \cong F/U$. This answers positively a conjecture of Lu and Yu from 2016.

math.AC↗

arXiv2023

On the notion of sequentially Cohen-Macaulay modules

In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help the reader with quickly getting acquainted with this topic. We then discuss two generalizations of the notion of sequential Cohen-Macaulayness which are inspired by a theorem of Jürgen Herzog and the third author.

math.AC↗

arXiv2023

Syzygies in Hilbert schemes of complete intersections

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$.

math.AC↗

arXiv2022

Explicit Stillman bounds for all degrees

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by n forms of degree at most d. Explicit values of the bounds for forms of degrees 5 and higher are not yet known. The main result of this article is the construction of explicit such bounds, for all degrees d, which behave like power towers of height d^3/6+11d/6-4. This is done by establishing a bound D(k,d), which controls the number of generators of a minimal prime over an ideal of a regular sequence of k or fewer forms of degree d, and supplementing it into Ananyan and Hochster's proof in order to obtain a recurrence relation.

math.AC↗

arXiv2022

The Eisenbud-Green-Harris Conjecture

We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature. We include a final section with some applications, and examples.

math.AC↗

arXiv2021

Bounds on the number of generators of prime ideals

Let $S$ be a polynomial ring over any field $\Bbbk$, and let $P \subseteq S$ be a non-degenerate homogeneous prime ideal of height $h$. When $\Bbbk$ is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in $P$ which only depends on $h$. We significantly extend this result by proving that the number of minimal generators of $P$ in any degree $j$ can be bounded above by an explicit function that only depends on $j$ and $h$. In addition to providing a bound for generators in any degree $j$, not just for quadrics, our techniques allow us to drop the assumption that $\Bbbk$ is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal.

math.AC↗

arXiv2021

Linearly presented modules and bounds on the Castelnuovo-Mumford regularity of ideals

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.

math.AC↗

arXiv2021

A hyperplane restriction theorem and applications to reductions of ideals

Green's general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over and infinite field K modulo a general linear form. We strengthen Green's result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O'Carroll on the Eakin-Sathaye theorem for complete and joint reductions.

math.AC↗

arXiv2020

The Eisenbud-Green-Harris conjecture for fast-growing degree sequences

Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence satisfy $d_i \geqslant \sum_{j=1}^{i-1}(d_j-1)$, improving a result obtained in 2008 by the first author and Maclagan. Except for the sporadic case of a regular sequence of five quadrics, recently proved by Güntürkün and Hochster, the results of this article recover all known cases of the conjecture where only the degrees of the regular sequence are fixed, and include several additional ones.

math.AC↗

arXiv2020

Decomposition of local cohomology tables of modules with large E-depth

We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich.

math.AC↗

arXiv2020

A Cayley-Bacharach theorem for points in $\mathbb{P}^n$

We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.

math.AG↗

arXiv2018

On the Lex-plus-powers Conjecture

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem.

math.AC↗

arXiv2018

Depth and regularity modulo a principal ideal

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

math.AC↗

arXiv2014

Poset Embeddings of Hilbert functions and Betti numbers

We study inequalities between graded Betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps, defined earlier by us in [Math. Z. 274, (2013), no. 3-4, pp. 809-819; arXiv:1009.4488]. We show that if graded Betti numbers do not decrease when we replace ideals in an algebra by their embedded versions, then the same behaviour is carried over to ring extensions. As a corollary we give alternative inductive proofs of earlier results of Bigatti, Hulett, Pardue, Mermin-Peeva-Stillman and Murai. We extend a hypersurface restriction theorem of Herzog-Popescu to the situation of embeddings. We show that we can obtain the Betti table of an ideal in the extension ring from the Betti table of its embedded version by a sequence of consecutive cancellations. We further show that the lex-plus-powers conjecture of Evans reduces to the Artinian situation.

math.AC↗