Algebraic surfaces as Hadamard products of curves

TL;DR

This paper investigates algebraic surfaces in projective 3-space formed as Hadamard products of curves, characterizing quadratic cases and exploring properties of higher-degree surfaces.

Contribution

It provides a classification of quadratic surfaces as Hadamard products of lines and analyzes the geometric properties of higher-degree cases.

Findings

Quadratic surfaces as Hadamard products of lines are smooth and tangent to coordinate planes.

Such quadratic surfaces are uniquely identified by tangency points.

Higher-degree surfaces as Hadamard products of a line and a curve have non-transversal intersections with coordinate planes.

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.