Determinantal varieties from point configurations on hypersurfaces

TL;DR

This paper studies the scheme parametrizing point configurations on hypersurfaces, revealing its determinantal structure and properties like irreducibility, Cohen-Macaulayness, and normality, along with a detailed analysis of its singular locus.

Contribution

It establishes the determinantal structure of the scheme $X_{r,d,n}$ and characterizes its singular locus using algebraic and geometric tools.

Findings

The scheme $X_{r,d,n}$ is irreducible, Cohen-Macaulay, and normal.

The singular locus is described via Castelnuovo-Mumford regularity and $d$-normality.

Characterizations are provided for specific cases $X_{2,d,n}$ and $X_{3,2,n}$.

Abstract

We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ 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 $X_{3,2,n}$.