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Emilia Mezzetti

Publications and source records attributed to Emilia Mezzetti.

At least 19 recordsLinked to original sources
arXiv2024

Perazzo $n$-folds and the weak Lefschetz property

In this paper, we determine the maximum $h_{max}$ and the minimum $h_{min}$ of the Hilbert vectors of Perazzo algebras $A_F$, where $F$ is a Perazzo polynomial of degree $d$ in $n+m+1$ variables. These algebras always fail the Strong Lefschetz Property. We determine the integers $n,m,d$ such that $h_{max}$ (resp. $h_{min}$) is unimodal, and we prove that $A_F$ always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in $\mathbb P^4$ with minimum Hilbert vectors. Finally we pose some open problems in this context. Dedicated to Enrique Arrondo on the occasion of his $60^{th}$ birthday.

math.AC↗
arXiv2023

Hilbert functions and Jordan type of Perazzo Artinian algebras

We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form.

math.AC↗
arXiv2023

Perazzo 3-folds and the weak Lefschetz property

We deal with Perazzo 3-folds in $\mathbb P^4$, i.e. hypersurfaces $X=V(f)\subset \mathbb P^4$ of degree $d$ defined by a homogeneous polynomial $f(x_0,x_1,x_2,u,v)=p_0(u,v)x_0+p_1(u,v)x_1+p_2(u,v)x_2+g(u,v)$, where $p_0,p_1,p_2$ are algebraically dependent but linearly independent forms of degree $d-1$ in $u,v$, and $g$ is a form in $u,v$ of degree $d$. Perazzo 3-folds have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra $A_f$ fails the strong Lefschetz property. In this paper, we determine the maximum and minimum Hilbert function of $A_f$ and we prove that if $A_f$ has maximal Hilbert function it fails the weak Lefschetz property, while it satisfies the weak Lefschetz property when it has minimum Hilbert function. In addition, we classify all Perazzo 3-folds in $\mathbb P^4$ such that $A_f$ has minimum Hilbert function.

math.AG↗
arXiv2022

Pencils of singular quadrics of constant rank and their orbits

We give a geometric description of singular pencils of quadrics of constant rank, relating them to the splitting type of some naturally associated vector bundles on $\mathbb{P}^1$. Then we study their orbits in the Grassmannian of lines, under the natural action of the general linear group.

math.AG↗
arXiv2021

Togliatti systems associated to the dihedral group and the weak Lefschetz property

In this note, we study Togliatti systems generated by invariants of the dihedral group $D_{2d}$ acting on $k[x_{0},x_{1},x_{2}]$. This leads to the first family of non monomial Togliatti systems, which we call $GT-$systems with group $D_{2d}$. We study their associated varieties $S_{D_{2d}}$, called $GT-$surfaces with group $D_{2d}$. We prove that they are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, $I(S_{D_{2d}})$, is minimally generated by quadrics and we find a minimal free resolution of $I(S_{D_{2d}})$.

math.AG↗
arXiv2020

On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems

Given any diagonal cyclic subgroup $Λ\subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ of degree $d$ which are invariants of $Λ$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case the projective toric variety $X_d$ parameterized by $(m_{1},\ldots, m_{r})$ is called a $GT$-variety with group $Λ$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.

math.AG↗
arXiv2020

Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$

We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle $E$ on the quadric. We analyse these bundles and their geometry, relating them to linear congruences of lines in $\mathbb{P}^5$.

math.AG↗
arXiv2018

On the Coefficients of the Permanent and the Determinant of a Circulant Matrix. Applications

Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.

math.AG↗
arXiv2018

Togliatti systems and Galois coverings

We study the homogeneous artinian ideals of the polynomial ring $K[x,y,z]$, generated by the homogenous polynomials of degree $d$ which are invariant under an action of the cyclic group $\mathbb Z/d\mathbb Z$, for any $d\geq 3$. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal $(1, e, e^a)$, where $e$ is a primitive $d$-th root of the unity. We get a complete description when $d$ is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.

math.AG↗
arXiv2018

Circulant matrices and Galois-Togliatti systems

The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.

math.AC↗
arXiv2017

Osculating behavior of Kummer surface in $\mathbb P^5$

In an article of 1967 W. Edge gave a description of some beautiful geometric properties of the Kummer surface complete intersection of three quadrics in $\mathbb P^5$. Working on it, R. Dye proved that all its osculating spaces have dimension less than the expected 5. Here we discuss these results, also at the light of some recent result about varieties with hypo-osculating behaviour.

math.AG↗
arXiv2017

Fano congruences of index $3$ and alternating $3$-forms

We study congruences of lines $X_ω$ defined by a sufficiently general choice of an alternating 3-form $ω$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$. We compute the degree of $X_ω$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $ω$ bijectively corresponds to $X_ω$ except when $n=5$. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for $n=8$ and the Peskine variety for $n=9$. The residual congruence $Y$ of $X_ω$ with respect to a general linear congruence containing $X_ω$ is analysed in terms of the quadrics containing the linear span of $X_ω$. We prove that $Y$ is Cohen-Macaulay but non-Gorenstein in codimension $4$. We also examine the fundamental locus $G$ of $Y$ of which we determine the singularities and the irreducible components.

math.AG↗
arXiv2016

Planes of matrices of constant rank and globally generated vector bundles

We consider the problem of determining all pairs (c_1, c_2) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c_1 and size 2c_1+2. We completely solve the problem in the "stable" range, i.e. for pairs with c_1^2-4c_2<0, proving that the additional condition c_2\le {{c_1+1}\choose 2} is necessary and sufficient. For c_1^2-4c_2\ge 0, we prove that there exist globally generated bundles, some even defining an embedding of P^2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c_1\le 3.

math.AG↗
arXiv2016

The minimal number of generators of a Togliatti system

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree $d$ in $n+1$ variables, for any $d\ge 2$ and $n\geq 2$. We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if $n=2$ (resp $n=2,3$) all range between the lower and upper bound is covered, while if $n\geq 3$ (resp. $n\ge 4$) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for $n=2$ the Mumford-Takemoto stability of the syzygy bundle associated to smooth monomial Togliatti systems.

math.AG↗
arXiv2013

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P^3 and derived category tools. The method allows one to prove the existence of new examples of size 10x10 and 14x14 via instanton bundles of charge 2 and 4 respectively, and provides an explanation for what used to be the only known example (Westwick 1996). We also give an algorithm to construct explicitly a matrix of size 14 of this type.

math.AG↗
arXiv2011

Laplace Equations and the Weak Lefschetz Property

We prove that r independent homogeneous polynomials of the same degree d become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose (d-1)-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.

math.AG↗
arXiv2010

Vector spaces of skew-symmetric matrices of constant rank

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2r and type (N+1)x(N+1) under the natural action of SL(N+1), over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1-generic matrices of linear forms. We also show that, for each rank two vector bundle on P^2 defining a triple Veronese embedding of P^2 in G(1,7), there exists a vector space of 8 x 8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.

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