arXiv subjects

Davide Bricalli

Publications and source records attributed to Davide Bricalli.

5 recordsLinked to original sources

arXiv2026

On the maximal variation problem and Lefschetz pencils

We study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard-Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of smooth projective varieties. This includes varieties $X$ of dimension $n\geq2$ with $p_g=h^{n,0}(X)>0$ and $H^{n-1,0}(X)=\{0\}$, Enriques surfaces, irregular surfaces with maximal Albanese dimension, smooth hyperk\"ahler varieties, and all the smooth not Fano hypersurfaces in $\mathbb{P}^n$. As a consequence, by a result of Beauville, we establish a Lefschetz property for the Jacobian rings of smooth hypersurfaces in $\mathbb{P}^n$ of degree n+1.

math.AG↗

arXiv2026

Linear spaces in Hessian loci of cubic hypersurfaces

In this paper we will study the Hessian hypersurface associated with a smooth cubic. We prove that the existence of a Hessian locus, associated with a smooth cubic form f, of dimension bigger then the expected one, forces the cubic f to be of Thom-Sebastiani type. Moreover, we will analyze the existence of some projective linear spaces in such Hessian loci and their nature in terms of the Hessian matrix. From this, we show that the only smooth cubic threefold having the same Hessian variety as the one associated with a general cubic form f of Waring Rank 6 is f itself. Finally, we prove that the hessian associated with a smooth hypersurface of any degree and dimension is not a cone.

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arXiv2024

On the irreducibility of Hessian loci of cubic hypersurfaces

We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if $f$ is not of Thom-Sebastiani type, i.e., roughly, one can not separate its variables. This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.

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arXiv2022

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of a weak Lefschetz property to an algebra from one of its quotients.

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arXiv2022

A theorem of Gordan and Noether via Gorenstein rings

Gordan and Noether proved in their fundamental theorem that an hypersurface $X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $n\geq 4$, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $\mathbb{K}$-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $R=\mathbb{K}[x_0,\dots,x_4]/J$ with $J$ generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.

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