arXiv subjects

Dario Antolini

Publications and source records attributed to Dario Antolini.

5 recordsLinked to original sources

arXiv2026

Finiteness of Hadamard ranks

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard rank is finite for any point. As a by-product we obtain the finiteness of the Hadamard rank with respect to varieties of tensors, such as Grassmannians, Chow varieties, varieties of reducible forms and their secant varieties, complementing previous known results on secant varieties of Segre-Veronese varieties. We prove sharp upper bounds on the maximum Hadamard rank for certain families of algebraic varieties: this is a consequence of a result on the lower semi-continuity of the Hadamard rank for curves that do not contain points with at least two zero coordinates.

math.AG↗

arXiv2025

Hadamard ranks of algebraic varieties

Motivated by the study of decompositions of tensors as Hadamard products (i.e., coefficient-wise products) of low-rank tensors, we introduce the notion of Hadamard rank of a given point with respect to a projective variety: if it exists, it is the smallest number of points in the variety such that the given point is equal to their Hadamard product. We prove that if the variety $X$ is not contained in a coordinate hyperplane or a binomial hypersurface, then the generic point has a finite $X$-Hadamard-rank. Although the Hadamard rank might not be well defined for special points, we prove that the general Hadamard rank with respect to secant varieties of toric varieties is finite and the maximum Hadamard rank for points with no coordinates equal to zero is at most twice the generic rank. In particular, we focus on Hadamard ranks with respect to secant varieties of toric varieties since they provide a geometric framework in which Hadamard decompositions of tensors can be interpreted. Finally, we give a lower bound to the dimension of Hadamard products of secant varieties of toric varieties: this allows us to deduce the general Hadamard rank with respect to secant varieties of several Segre-Veronese varieties.

math.AG↗

arXiv2025

Dilworth truncations and Hadamard products of linear spaces

As a direct application of Dilworth truncations of polymatroids, we give short proofs of two theorems: Bernstein's characterisation of algebraic matroids coming from the Hadamard product of two linear spaces, and a formula for the dimension of the amoeba of a complex linear space by Draisma, Eggleston, Pendavingh, Rau, and Yuen. We disprove Bernstein's conjecture on a characterisation of the algebraic matroids of Hadamard products of more than two linear spaces, by giving explicit counterexamples.

math.CO↗

arXiv2025

Algebraic surfaces as Hadamard products of curves

We study projective surfaces in $\mathbb{P}^3$ which can be written as Hadamard product of two curves. We show that quadratic surfaces which are Hadamard product of two lines are smooth and tangent to all coordinate planes, and such tangency points uniquely identify the quadric. The variety of such quadratic surfaces corresponds to the Zariski closure of the space of symmetric matrices whose inverse has null diagonal. For higher-degree surfaces which are Hadamard product of a line and a curve we show that the intersection with the coordinate planes is always non-transversal.

math.AG↗

arXiv2024

The Chirotropical Grassmannian

Recent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian $\text{Trop}^\chi \text{G}(k,n)$ and the chirotropical Dressian $\text{Dr}^\chi(k,n)$, polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that $\text{Trop}^\chi \text{G}(3,n) = \text{Dr}^\chi(3,n)$ for $n = 6,7,8$, and develop algorithms to compute these objects from their rays modulo lineality. Using these algorithms, we compute all chirotropical Grassmannians $\text{Trop}^\chi \text{G}(3,n)$ for $n = 6,7,8$ across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space $X^\chi(3,6)$ is diffeomorphic to a polytope and propose an associated canonical logarithmic differential form. Finally, we show that the equality between chirotropical Grassmannian and Dressian fails for $(k,n) = (4,8)$.

math.CO↗