Some Taylor varieties with null Hessian

TL;DR

This paper investigates Taylor varieties from rational function expansions, identifying new hypersurfaces with zero Hessian in specific cases, contributing to algebraic geometry and tensor analysis.

Contribution

It proves that certain Taylor varieties with specific parameters produce new examples of hypersurfaces with null Hessian, expanding understanding of these geometric objects.

Findings

Identifies new hypersurfaces with null Hessian for n=2 and m=d+2.

Establishes conditions under which Taylor varieties have null Hessian.

Contributes to the classification of defective hypersurfaces in algebraic geometry.

Abstract

Taylor varieties $\mathcal{T}^n_{d,e,m}$ arise from Taylor expansion of rational functions in $n$ variables. Among them, we look for non-defective hypersurfaces. We prove that the cases $n=2$ and $m=d+2$ give new examples of hypersurfaces with identically null Hessian.