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Elena Guardo

Publications and source records attributed to Elena Guardo.

At least 19 recordsLinked to original sources
arXiv2026

Geometrically vertex decomposable star configurations

The goal of this paper is to determine how the family of ideals of star configurations intersects with the class of geometrically vertex decomposable ideals. The main result of this paper shows that the answer is subtle since the geometrically vertex decomposability property of an ideal is not invariant under a linear change of variables, and thus the answer will depend upon the choice of the linear forms that define the ideal of the star configuration. We also show that the ideal of a star configuration is a Knutson ideal precisely when it is a geometrically vertex decomposable ideal.

math.AC↗
arXiv2026

Some Taylor varieties with null Hessian

Taylor varieties $\mathcal{T}^n_{d,e,m}$ arise from Taylor expansion of rational functions in $n$ variables. Among them, we look for non-defective hypersurfaces. We prove that the cases $n=2$ and $m=d+2$ give new examples of hypersurfaces with identically null Hessian.

math.AG↗
arXiv2026

Interpolation matrices and jumping lines of logarithmic bundles

We study jumping lines loci of logarithmic bundles associated with finite sets of points in the projective plane. Using the interpolation matrix introduced in [DMTG25], we describe these loci as the zero sets of explicit determinants depending on parameters $(d,m)$ determined by the number of points. We show that for points in general position the determinant defines an irreducible curve of the expected degree, while for special configurations it acquires fixed components related to the combinatorics of the arrangement. The approach provides a new geometric interpretation of the classical jumping lines of Dolgachev--Kapranov and Barth, and connects them to the framework of unexpected curves and hypersurfaces.

math.AG↗
arXiv2025

Splittings of Ideals of Points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$

Let $I_\mathbb{X}$ be the bihomogeneous ideal of a finite set of points $\mathbb{X} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$. The purpose of this note is to consider ``splittings'' of the ideal $I_\mathbb{X}$, that is, finding ideals $J$ and $K$ such that $I_\mathbb{X} = J+K$, where $J$ and $K$ have prescribed algebraic or geometric properties. We show that for any set of points $\mathbb{X}$, we cannot partition the generators of $I_\mathbb{X}$ into two ideals of points. The best case scenario is where at most one of $J$ or $K$ is an ideal of points. To remedy this we introduce the notion of unions of lines and ACM (Arithmetically Cohen-Macaulay) points which allows us to say more about splittings. For a set $\mathbb{W}$ of unions of lines and ACM sets of points, we can write $I_\mathbb{W} = J + K$ where both $J$ and $K$ are ideals of unions of lines and ACM points as well. When $\mathbb{W}$ is a union of lines and ACM points, we discuss some consequences for the graded Betti numbers of $I_{\mathbb{W}}$ in terms of these splittings.

math.AC↗
arXiv2024

On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

Published version: We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially $(\mathbb P^1)^n$. A combinatorial characterization, the $(\star)$-property, is known in $\mathbb P^1 \times \mathbb P^1$. We propose a combinatorial property, $(\star_n)$, that directly generalizes the $(\star)$-property to $(\mathbb P^1)^n$ for larger $n$. We show that $X$ is ACM if and only if it satisfies the $(\star_n)$-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space. Corrigendum: We correct a mistake in the cited paper. It introduced a combinatorial property, the $(\star_n)$-property, for a finite set of points $X$ in $(\mathbb P^1)^n$ and claimed that this property holds if and only if $X$ is ACM. In fact $X$ being ACM is a sufficient condition for the $(\star_n)$-property, but we only prove that it is necessary when $n=3$, and we give a counterexample when $n=4$.

math.AG↗
arXiv2023

The Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$

A $\Bbbk$-configuration of type $(d_1,\dots,d_s)$ is a specific set of points in $\mathbb P^2$ that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all $\Bbbk$-configurations in $\mathbb P^2$ are determined by the type $(d_1,\dots,d_s)$. However the Waldschmidt constant of a $\Bbbk$-configuration in $\mathbb P^2$ of the same type may vary. In this paper, we find that the Waldschmidt constant of a $\Bbbk$-configuration in $\mathbb P^2$ of type $(d_1,\dots,d_s)$ with $d_1\ge s\ge 1$ is $s$. We also find the Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of type $(a,b,c)$ with $a\ge 1$ except the type $(2,3,5)$. In particular, we prove that the Waldschmidt constant of a standard $\Bbbk$-configuration in $\mathbb P^2$ of type $(1,b,c)$ with $c\ge 2b+2$ does not depend on $c$.

math.AG↗
arXiv2021

Finite $0$-dimensional multiprojective schemes and their ideals

We study finite $0$-dimensional schemes in product of multiprojective spaces and their ideals. In particular, we describe the set of generators of the ideal defining a $0$-dimensional scheme in the case $\mathbb P^{1}\times\cdots \times\mathbb P^{1}$ and in the case $\mathbb P^{n_1}\times \cdots \times \mathbb P^{n_k}$ with $n_i\in \{1,2\}$ for all $i$. We also check very ampleness for zero-dimensional schemes of general points in the multiprojective spaces.

math.AG↗
arXiv2021

Rational normal curves and Hadamard products

Given $r>n$ general hyperplanes in $\mathbb P^n,$ a star configuration of points is the set of all the $n$-wise intersection of them. We introduce {\it contact star configurations}, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in $\mathbb P^2$, we prove that the union of two contact star configurations has a special $h$-vector and, in some cases, this is a complete intersection.

math.AG↗
arXiv2021

Steiner Configurations ideals: containment and colouring

Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -- Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathbb{P}^{n}_{k}$. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly's Conjectures. Moreover, given a hypergraph $H$, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to $H$. We apply these results in the case that $H$ is a Steiner System.

math.AC↗
arXiv2020

Steiner systems and configurations of points

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

math.AG↗
arXiv2020

Symbolic powers of codimension two Cohen-Macaulay ideals

Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in $\mathbb{P}^1 \times \mathbb{P}^1$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

math.AC↗
arXiv2020

Complex uniformly resolvable decompositions of $K_v$

In this paper we consider the complex uniformly resolvable decompositions of the complete graph $K_v$ into subgraphs such that each resolution class contains only blocks isomorphic to the same graph from a given set $\mathcal H$. We completely determine the spectrum for the cases $\mathcal{H} = \{K_2, P_3, K_3\}$, $\mathcal{H} = \{P_4, C_4\}$, and $\mathcal{H} = \{K_2, P_4, C_4\}$.

math.CO↗
arXiv2020

Expecting the unexpected: quantifying the persistence of unexpected hypersurfaces

If $X \subset \mathbb P^n$ is a reduced subscheme, we say that $X$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$ if the imposition of having multiplicity $m$ at a general point $P$ fails to impose the expected number of conditions on the linear system of hypersurfaces of degree $t$ containing $X$. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understand. We introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of $X$ which in some cases guarantee and in other cases preclude $X$ having certain kinds of unexpectedness. In addition, we formulate a new way of quantifying unexpectedness (our AV sequence), which allows us detect the extent to which unexpectedness persists as $t$ increases but $t-m$ remains constant. Finally, we study to what extent we can detect unexpectedness from the Hilbert function of $X$.

math.AG↗
arXiv2019

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq \mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that $H$ is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.

math.AC↗
arXiv2018

Kaehler differentials for fat point schemes in P^1xP^1

Let $X$ be a set of $K$-rational points in $P^1 \times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $Ω^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 \times P^1$ having the Cayley-Bacharach property.

math.AG↗
arXiv2018

Hadamard Star Configurations

Bocci, Carlini, and Kileel have shown that the square-free Hadamard product of a finite set of points $Z$ that all lie on a line $\ell$ in $\mathbb{P}^n$ produces a star configuration of codimension $n$. In this paper we introduce a construction using the Hadamard product to construct star configurations of codimension $c$. In the case that $c = n= 2$, our construction produces the star configurations of Bocci, Carlini, and Kileel. We will call any star configuration that can be constructed using our approach a Hadamard star configuration. Our main result is a classification of Hadamard star configurations.

math.AG↗