arXiv subjects

Luca Sodomaco

Publications and source records attributed to Luca Sodomaco.

21 records · Page 2Linked to original sources

arXiv2019

On the unique unexpected quartic in $\mathbb{P}^2$

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently Di Gennaro, Ilardi and Vallès discovered a special configuration $Z$ of nine points with a remarkable property: a general triple point always fails to impose independent conditions on the ideal of $Z$ in degree four. The peculiar structure and properties of this kind of \textit{unexpected curves} were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low degree unexpected curves. In particular, we prove that the aforementioned configuration $Z$ is the unique one giving rise to an unexpected quartic.

math.AG↗

arXiv2019

On the product of the singular values of a binary tensor

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $μ\vdash d$ and a $μ$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{μ,\mathbb{R}}$ of $μ$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_μ$ the complexification of $X_{μ,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ of the dual variety of $X_μ$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ is the square of the $μ$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $μ$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ for tensors of format $2\times 2\times 2$.

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arXiv2018

The product of the eigenvalues of a symmetric tensor

We study E-eigenvalues of a symmetric tensor $f$ of degree $d$ on a finite-dimensional Euclidean vector space $V$, and their relation with the E-characteristic polynomial of $f$. We show that the leading coefficient of the E-characteristic polynomial of $f$, when it has maximum degree, is the $(d-2)$-th power (respectively the $((d-2)/2)$-th power) when $d$ is odd (respectively when $d$ is even) of the $\widetilde{Q}$-discriminant, where $\widetilde{Q}$ is the $d$-th Veronese embedding of the isotropic quadric $Q\subseteq\mathbb{P}(V)$. This fact, together with a known formula for the constant term of the E-characteristic polynomial of $f$, leads to a closed formula for the product of the E-eigenvalues of $f$, which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues.

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