On rank 3 quadratic equations of Veronese varieties

TL;DR

This paper investigates the geometric structure and irreducible decomposition of the locus of rank 3 quadratic equations associated with Veronese varieties, analyzing their singularities and desingularizations.

Contribution

It provides a detailed analysis of the irreducible components of the rank 3 quadratic equations locus of Veronese varieties, including their geometric properties and singularity structure.

Findings

Identified the minimal irreducible decomposition of the locus

Analyzed the desingularizations of irreducible components

Determined the singularity and non-singularity conditions

Abstract

This paper studies the geometric structure of the locus $\Phi_3 (X)$ of rank $3$ quadratic equations of the Veronese variety $X = \nu_d (\mathbb{P}^n)$. Specifically, we investigate the minimal irreducible decomposition of $\Phi_3 (X)$ of rank $3$ quadratic equations and analyze the geometric properties of the irreducible components of $\Phi_3 (X)$ such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of $\Phi_3 (X)$.