On rank 3 quadratic equations of Veronese varieties

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 $\Phi_3 (X)$.