On rank 3 quadratic equations of Veronese varieties

This paper studies the geometric structure of the locus ${\mathrm{\Phi }}_3(X)$ of rank $3$ quadratic equations of the Veronese variety $X={\nu }_d({\mathbb{P}}^n)$. Specifically, we investigate the minimal irreducible decomposition of ${\mathrm{\Phi }}_3(X)$ of rank $3$ quadratic equations and analyze the geometric properties of the irreducible components of ${\mathrm{\Phi }}_3(X)$ such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of ${\mathrm{\Phi }}_3(X)$.