On rank 3 quadratic equations of Veronese embeddings

초록

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

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