arXiv subjects

Maria Virginia Catalisano

Publications and source records attributed to Maria Virginia Catalisano.

At least 19 recordsLinked to original sources

arXiv2025

Postulation for 2-superfat points in the plane

We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes have good postulation, i.e., they have the expected Hilbert function. We also show the good postulation of such schemes when we add a general 3-fat point. Finally, we use these results to answer a peculiar kind of interpolation problem.

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arXiv2024

Superfat points and associated tensors

We consider 0-dimensional schemes supported at a single point in n-space that are m-symmetric, i.e. that intersect any smooth curve passing through the point with length m, and the ones among them that are maximal with respect to inclusion (called m-superfat points). We study properties of such schemes, in particular for n=2. We give a first application of the simplest such schemes, namely 2-superfat points in the plane, by studying varieties defined by them on Veronese and Segre-Veronese varieties and the (symmetric or partially symmetric) tensors they parameterize.

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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 $c$.

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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.

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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.

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arXiv2018

The Hitchhiker guide to: Secant Varieties and Tensor Decomposition

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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arXiv2018

Tangential varieties of Segre-Veronese surfaces are never defective

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.

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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.

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arXiv2017

On the Hilbert function of general fat points in $\mathbb{P}^1 \times \mathbb{P}^1$

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.

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arXiv2017

On the Hartshorne-Hirschowitz theorem

The Hartshorne--Hirschowitz theorem says that a generic union of lines in $\mathbb{P}^n$, $(n\geq 3)$, has good postulation. The proof of Hartshorne and Hirschowitz in the initial case $\mathbb{P}^3$ is difficult and so long, which is handled by a method of specialization via a smooth quadric surface with the property of having two rulings of skew lines. We provide a proof in the case $\mathbb{P}^3$ based on a new degeneration of disjoint lines via a plane $H\cong\mathbb{P}^2$, which we call $(2,s)$-cone configuration, that is a schematic union of $s$ intersecting lines passing through a single point $P$ together with the trace of an $s$-multiple point supported at $P$ on the double plane $2H$. In the first part of this paper, we discuss our degeneration inductive approach. We prove that a $(2,s)$-cone configuration is a degeneration of $s$ disjoint lines in $\mathbb{P}^3$, or more generally in $\mathbb{P}^n$. In the second part of the paper, we use this degeneration in an effective method to show that a generic union of lines in $\mathbb{P}^3$ imposes independent conditions on the linear system $|\mathbb{O}_{\mathbb{P}^3}(d)|$ of surfaces of given degree $d$. The basic motivation behind our degeneration approach is that it looks more systematic that gives some hope of extensions to the analogous problem in higher dimensional spaces, that is the postulation problem for $m$-dimensional planes in $\mathbb{P}^{2m+1}$.

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arXiv2017

Waring-like decompositions of polynomials - 1

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.

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arXiv2017

Waring loci and the Strassen conjecture

The Waring locus of a form F is the collection of the degree one forms appearing in some minimal sum of powers decomposition of F. In this paper, we give a complete description of Waring loci for several family of forms, such as quadrics, monomials, binary forms and plane cubics. We also introduce a Waring loci version of Strassen's Conjecture, which implies the original conjecture, and we prove it in many cases.

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arXiv2015

Symmetric tensors: rank and Strassen's conjecture

In this paper we introduce the notion of linear computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

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arXiv2015

$e$-computable forms and the Strassen conjecture

In this paper we introduce the notion of $e$-computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

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arXiv2014

The secant line variety to the varieties of reducible plane curves

Let $λ=[d_1,\dots,d_r]$ be a partition of $d$. Consider the variety $\mathbb{X}_{2,λ} \subset \mathbb{P}^N$, $N={d+2 \choose 2}-1$, parameterizing forms $F\in k[x_0,x_1,x_2]_d$ which are the product of $r\geq 2$ forms $F_1,\dots,F_r$, with deg$F_i = d_i$. We study the secant line variety $σ_2(\mathbb{X}_{2,λ})$, and we determine, for all $r$ and $d$, whether or not such a secant variety is defective. Defectivity occurs in infinitely many "unbalanced" cases.

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arXiv2014

Progress on the symmetric Strassen conjecture

Let F and G be homogeneous polynomials in disjoint sets of variables. We prove that the Waring rank is additive, thus proving the symmetric Strassen conjecture, when either F or G is a power, or F and G have two variables, or either F or G has small rank.

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arXiv2013

Reduced and non-reduced linear spaces: Lines and points

In this paper we consider the problem of determining the Hilbert function of schemes X of the proiective space P^n which are the generic union of s lines and one m-multiple point. We completely solve this problem for any s and m when n > 3. When n=3 we find several defective such schemes and conjecture that they are the only ones. We verify this conjecture in several cases.

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arXiv2012

Grassmann secants, identifiability, and linear systems of tensors

For any irreducible non-degenerate variety $X\subset \mathbb{P}^r$, we give a criterion for the $(k,s)$-identifiability of $X$. If $k\leq s-1 <r$, then the $(k,s)$-identifiability holds for $X$ if and only if the $s$-identifiability holds for the Segre product $Seg(\mathbb{P}^k\times X)$. Moreover, if the $s$-th secant variety of $X$ is not defective and it does not fill the ambient space, then we can produce a family of pairs $(k,s)$ for which the $(k,s)$-identifiability holds for $X$.

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