On singularities of third secant varieties of Veronese embeddings

TL;DR

This paper investigates the singularity structure of the third secant varieties of Veronese embeddings, which relate to symmetric tensors with border rank up to three, enhancing understanding of their geometric and algebraic properties.

Contribution

It provides a detailed analysis of the singularities of third secant varieties of Veronese embeddings, a topic not extensively explored before.

Findings

Identification of singular loci in third secant varieties

Characterization of singularities in relation to tensor border rank

Insights into the geometric structure of symmetric tensor spaces

Abstract

In this paper we study singularities of third secant varieties of Veronese embedding $v_d(\mathbb{P}^n)$, which corresponds to the variety of symmetric tensors of border rank at most three in $(\mathbb{C}^{n+1})^{\otimes d}$.