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Valentina Beorchia

Publications and source records attributed to Valentina Beorchia.

18 recordsLinked to original sources
arXiv2025

Free plane curves with a linear Jacobian syzygy

The study of planar free curves is a very active area of research, but a structural study of such a class is missing. We give a complete classification of the possible generators of the Jacobian syzygy module of a plane free curve under the assumption that one of them is linear. Specifically, we prove that, up to similarities, there are two possible forms for the Hilbert-Burch matrix. Our strategy relies on a translation of the problem into the accurate study of the geometry of maximal segments of a suitable triangle with integer points. Following this description, we are able to determine precisely the equations of free curves and the associated Hilbert-Burch matrices.

math.AC↗
arXiv2025

Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the identification of quasi-homogeneous hypersurface isolated singularities. The result gives an insight on how the geometry is reflected in the Jacobian syzygies and extends previous results of the first, second and last author for free and nearly free plane curves [1].

math.AG↗
arXiv2025

Weak Lefschetz property of equigenerated complete intersections. Applications

In this paper, we prove that any Artinian complete intersection homogeneous ideal $I$ in $K[x_0,\cdots,x_n]$ generated by $n+1$ forms of degree $d\ge 2$ satisfies the weak Lefschetz property (WLP) in degree $t< d+\lceil \frac{d}{n} \rceil$. As a consequence, we get that the Jacobian ideal of a smooth 3-fold of degree $d\ge 7$ in ${\mathbb P}^4$ satisfies the weak Lefschetz property in degree $d$, answering a recent question of Beauville.

math.AG↗
arXiv2025

On the generic injectivity of Hessian maps of ternary forms

We study the problem of the generic injectivity of the Hessian map, associating with a proportionality class of a ternary form the class of its Hessian determinant, conjectured by C. Ciliberto and G. Ottaviani and recently proved by the same authors. Taking into account that the Hessian curve is the ramification divisor associated with the polar map, we perform a study of the problem using a geometric description of the graph of such a map.

math.AG↗
arXiv2024

Eigenpoint collinearities of plane cubics

Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or more alignments. We give a classification and explicit equations, in the coordinates of the points, of all configurations: this is accomplished by using both geometric techniques and by an extensive use of computer algebra.

math.AG↗
arXiv2023

Jacobian schemes of conic-line arrangements and eigenschemes

The Jacobian scheme of a reduced, singular projective plane curve is the zero-dimensional scheme, whose homogeneous ideal is generated by the partials of its defining polynomial. The degree of such a scheme is called the global Tjurina number and, if the curve is not a set of concurrent lines, some upper and lower bounds depending on the degree of the curve and the minimal degree of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall. In this paper we give a complete geometric characterization of conic-line arrangenents, with global Tjurina number attaining the upper bound. Furthermore, we characterize conic-line arrangenents attaining the lower bound for the global Tjurina number, among all curves with a linear Jacobian syzygy. As an application, we characterize conic-line arrangenents with Jacobian scheme equal to an eigenscheme of some ternary tensor, and we study the geometry of their polar maps.

math.AG↗
arXiv2022

On the slope inequalities for extremal curves

The present paper concerns the question of the violation of the r-th inequality for extremal curves in the projective r-space, posed by T. Kato and G. Martens. We show that the answer is negative in many cases. The result is obtained by a detailed analysis of the geometry of extremal curves and their canonical model. As a consequence, we show that particular curves on a Hirzebruch surface do not violate the slope inequalities in a certain range.

math.AG↗
arXiv2022

Configurations of eigenpoints

This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is the complete intersection of two suitable smooth determinantal curves on a smooth determinantal surface. Moreover, we prove that the converse result holds if n=3, providing an answer in any degree to the cited question. Finally, we show that any general set of points in the projective 3-space can be enlarged to an eigenscheme of a partially symmetric tensor.

math.AG↗
arXiv2022

Generic identifiability of pairs of ternary forms

We prove that two general ternary forms are simultaneously identifiable only in the classical cases of two quadratic and a cubic and a quadratic form. We translate the problem into the study of a certain linear system on a projective bundle on the plane, and we apply techniques from projective and birational geometry to prove that the associated map is not birational.

math.AG↗
arXiv2022

Equations of tensor eigenschemes

We study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme.

math.AG↗
arXiv2021

Eigenschemes of Ternary Tensors

We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely both on classical and modern complex projective algebraic geometry.

math.AG↗
arXiv2018

Trigonal deformations of rank one and Jacobians

In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Jacobian J(C) is dominated by a hyperelliptic Jacobian of genus g' greater or equal to g.

math.AG↗
arXiv2018

The maximum genus problem for locally Cohen-Macaulay space curves

Let $P_{\text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{\text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. With the aid of \emph{Macaulay2} we checked the validity of the conjecture for $s \leq 100$. From the conjecture it would follow that $P(d,s)= P_{\text{MAX}}(d,s)$ for $d=s$ and for every $d \geq 2s-1$.

math.AG↗
arXiv2018

Generically nef vector bundles on ruled surfaces

The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor. Finally, we use such results to bound the invariants of curve fibrations, which factorize through finite covers of ruled surfaces.

math.AG↗
arXiv2017

On the slope of fourgonal semistable fibrations

We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.

math.AG↗
arXiv2013

A note on Harris Morrison sweeping families of maximal gonality

Harris and Morrison constructed semistable families f:F \to Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio K_F^2 / χ(O_F) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F\to Y is 12 asymptotically with respect to g and that F is a surface of positive index.

math.AG↗
arXiv2004

Canonical map of low codimensional subvarieties

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq λd^{ε_1}+c(\sum_{1\leq h < ε_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ of dimension $h$, and where $λ$, $ε_1$ and $ε_2$ are suitable positive real numbers depending only on the dimension of $X$, on $a$ and on the ambient variety $V$. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety $X$ of $V$, is birational.

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