Determinantal Varieties From Point Configurations on Hypersurfaces
Alessio Caminata,
Han-Bom Moon,
Luca Schaffler
Abstract:We consider the scheme $X_{r,d,n}$ parameterizing $n$ 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}… Show more
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