Algebraic surfaces as Hadamard products of curves

TL;DR

The paper addresses when a projective surface in $\mathbb{P}^3$ can be expressed as a Hadamard product of two curves, with a primary focus on quadratic surfaces that arise as the Hadamard product of two lines. It develops the Hadamard-product framework, proves that the resulting quadrics are smooth and tangent to all coordinate planes, and identifies a 5-dimensional moduli space of such quadrics, parametrized by symmetric $4\times4$ matrices with null diagonal via $A \mapsto A^{-1}$. It establishes identifiability: the four singular points of the intersections with the coordinate planes uniquely determine the quadric. The work also extends to higher-degree cases, showing that line–curve Hadamard products yield non-transversal intersections with coordinate planes when the product is a morphism, and that higher-degree products typically produce cones or singularities along curves, with examples illustrating the limitations and structure. Overall, the results provide concrete algebraic and geometric criteria for when a surface decomposes as a Hadamard product of curves, with potential implications for algebraic statistics and tensor geometry. $

Abstract

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.

Theorems & Definitions (32)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 3.1
• Lemma 3.2
• proof
• proof : Proof of \ref{['theorem:quadraticsurfaces']}
• Claim