Determinantal varieties from point configurations on hypersurfaces

2022/11/23 by A. Caminata, Caminata, Alessio, Han‐Bom Moon +3 · 1 citation
Mathematics · #Commutative Algebra and Its Applications #Tensor decomposition and applications #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2211.13177

Abstract

We consider the scheme X_r,d,n parametrizing n ordered points in projective space ℙ^r that lie on a common hypersurface of degree d. We show that this scheme has a determinantal structure and we prove that it is irreducible, Cohen-Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of X_r,d,n in terms of Castelnuovo-Mumford regularity and d-normality. This yields a characterization of the singular locus of X_2,d,n and X_3,2,n.

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