arXiv subjects

Alessio Sammartano

Publications and source records attributed to Alessio Sammartano.

At least 19 recordsLinked to original sources

arXiv2026

The structure of almost symmetric almost complete intersection numerical semigroups

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. Specifically, we show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the associated semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Moreover, both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, we show that this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that it is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several additional key results. 1) A rigidity theorem: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result can be regarded as the ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 3) Herzog--Watanabe question: en route to proving the main theorem, we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

math.AC↗

arXiv2026

Cartwright-Sturmfels Hilbert Schemes

Let S be the Cox ring of a product of r projective spaces. In this paper, we study the Cartwright-Sturmfels Hilbert schemes of S, which are multigraded Hilbert schemes that only parametrize radical ideals. Our main result shows that these Hilbert schemes are always smooth and irreducible if the Picard rank r is at most 2. This result can be seen as a multigraded analogue of the famous theorems of Fogarty and Maclagan-Smith, where the Picard rank replaces the dimension of the ambient space.

math.AG↗

arXiv2026

Irrational components of the Hilbert scheme of points

We construct irrational irreducible components of the Hilbert scheme of points of affine n-dimensional space, for n at least 12. We start with irrational components of the Hilbert scheme of curves in P^3 and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of A^n. The result solves Problem XX of [J. Jelisiejew, Open problems in deformations of Artinian algebras, Hilbert schemes and around, arXiv:2307.08777, 2023].

math.AG↗

arXiv2026

Behrend function and blowup algebras

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $ν_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson--Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo--Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.

math.AC↗

arXiv2026

Open problems on relations of numerical semigroups

We collect some open problems about minimal presentations of numerical semigroups and, more generally, about defining ideals and free resolutions of their semigroup rings and associated graded rings. We emphasize both long-standing problems and more recent questions and developments.

math.CO↗

arXiv2026

The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture that it is exhaustive: every smooth point admits a broken Gorenstein structure. We give an explicit characterization of the smooth points on the Hilbert scheme of A^3 corresponding to monomial ideals. We investigate the nature of the singular points, and prove several conjectures by Hu. Along the way, we obtain a number of additional results, related to linkage classes, nested Hilbert schemes, and a bundle on the Hilbert scheme of a surface.

math.AG↗

arXiv2025

The variety of orthogonal frames

An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We classify the irreducible components of V(d,n), give criteria for the ideal I(d,n) to be prime or a complete intersection, and for the variety V(d,n) to be normal. We also give near-equivalent conditions for V(d,n) to be factorial. Applications are given to the theory of Lovász-Saks-Schrijver ideals.

math.AC↗

arXiv2025

Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.

math.AC↗

arXiv2024

The geometry of double nested Hilbert schemes of points on curves

Let $C$ be a smooth curve. In this paper we investigate the geometric properties of the double nested Hilbert scheme of points on $C$, a moduli space introduced by the third author in the context of BPS invariants of local curves and sheaf counting on Calabi-Yau 3-folds. We prove this moduli space is connected, reduced and of pure dimension; we list its components via an explicit combinatorial characterisation and we show they can be resolved, when singular, by products of symmetric products of $C$. We achieve this via a purely algebraic analysis of the factorisation properties of the monoid of reverse plane partitions. We discuss the (virtual) fundamental class of the moduli space, we describe the local equations cutting it inside a smooth ambient space, and finally we provide a closed formula for its motivic class in the Grothendieck ring of varieties.

math.AG↗

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↗

arXiv2024

On minimal presentations of numerical monoids

We consider the classical problem of determining the largest possible cardinality of a minimal presentation of a numerical monoid with given embedding dimension and multiplicity. Very few values of this cardinality are known. In addressing this problem, we apply tools from Hilbert functions and free resolutions of artinian standard graded algebras. This approach allows us to solve the problem in many cases and, at the same time, identify subtle difficulties in the remaining cases. As a by-product of our analysis, we deduce results for the corresponding problem for the type of a numerical monoid.

math.CO↗

arXiv2024

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle Δ\rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $Δ$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

math.AC↗

arXiv2023

On the parity conjecture for Hilbert schemes of points on threefolds

Let $Hilb^d(A^3)$ be the Hilbert scheme of $d$ points in $A^3$, and let $T_z$ denote the tangent space to a point $z \in Hilb^d(A^3)$. Okounkov and Pandharipande have conjectured that $\dim T_z$ and $d$ have the same parity for every $z$. For points $z$ parametrizing monomial ideals, the conjecture was proved by Maulik, Nekrasov, Okounkov, and Pandharipande. In this paper, we settle the conjecture for points $z$ parametrizing homogeneous ideals. In fact, we state a generalization of the conjecture to Quot schemes of $A^3$, and we prove it for points parametrizing graded modules.

math.AG↗

arXiv2023

On Rees algebras of 2-determinantal ideals

Let I be the ideal of minors of a 2 by n matrix of linear forms with the expected codimension. In this paper we prove that the Rees algebra of I and its special fiber ring are Cohen-Macaulay and Koszul; in particular, they are quadratic algebras. The main novelty in our approach is the analysis of a stratification of the Hilbert scheme of determinantal ideals. We study degenerations of Rees algebras along this stratification, and combine it with certain squarefree Groebner degenerations.

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

Rational singularities of nested Hilbert schemes

The Hilbert scheme of points $\mathrm{Hilb}^n(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\mathrm{Hilb}^n(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\mathrm{Hilb}^n(S)$, the nested Hilbert scheme $\mathrm{Hilb}^{(n_1,n_2)}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_1 \supseteq Z_2$ of $S$ with $\mathrm{deg} (Z_i)=n_i$. In contrast to Fogarty's theorem, $\mathrm{Hilb}^{(n_1,n_2)}(S)$ is almost always singular, and very little is known about its singularities. In this paper we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70's shows that $\mathrm{Hilb}^{(n,1)}(S)$ is a normal Cohen-Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\mathrm{Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\mathrm{Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including square-free Gröbner degenerations, the Stanley-Reisner correspondence, and the Kempf-Lascoux-Weyman technique of calculating syzygies. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

math.AC↗

arXiv2022

On the tangent space to the Hilbert scheme of points in P3

In this paper we study the tangent space to the Hilbert scheme $\mathrm{Hilb}^d \mathbf{P}^3$, motivated by Haiman's work on $\mathrm{Hilb}^d \mathbf{P}^2$ and by a long-standing conjecture of Briançon and Iarrobino on the most singular point in $\mathrm{Hilb}^d \mathbf{P}^n$. For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of 4/3, and improve the known asymptotic bound on the dimension of $\mathrm{Hilb}^d \mathbf{P}^3$. Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.

math.AG↗

arXiv2021

On the smoothness of lexicographic points on Hilbert schemes

We study the geometry of standard graded Hilbert schemes of polynomial rings and exterior algebras. Our investigation is motivated by a famous theorem of Reeves and Stillman for the Grothendieck Hilbert scheme, which states that the lexicographic point is smooth. By contrast, we show that, in standard graded Hilbert schemes of polynomial rings and exterior algebras, the lexicographic point can be singular, and it can lie in multiple irreducible components. We answer questions of Peeva-Stillman and of Maclagan-Smith.

math.AG↗