On rank 3 quadratic equations of Veronese embeddings

Abstract This paper studies the geometric structure of the locus of rank 3 quadratic equations of the Veronese variety . Specifically, we investigate the minimal irreducible decomposition of of rank 3 quadratic equations and analyze the geometric properties of the irreducible components of such as their desingularizations. Additionally, we explore the non‐singularity and singularity of these irreducible components of .