Projective surfaces not as Hadamard products and the dimensions of the Hadamard joins

TL;DR

This paper investigates the dimensions of Hadamard products of multiple varieties under projective transformations and establishes conditions under which the join of a variety and a curve has the expected dimension.

Contribution

It introduces new results on the dimension behavior of Hadamard products with transformations and characterizes the dimension of joins involving certain varieties.

Findings

Hadamard product dimensions can be controlled via projective transformations.

Join of a non-hyperplane-contained variety with a 'nice' curve has expected dimension.

Provides new insights into the geometric properties of Hadamard products and joins.

Abstract

We study the dimensions of Hadamard products of $k\ge 3$ varieties if we allow to modify $k-1$ of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate hyperplane with a ``nice'' curve always has the expected dimension.