Determinantal Varieties From Point Configurations on Hypersurfaces

We consider the scheme X-r,X-d,X-n parameterizing 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,X-d,X-n in terms of Castelnuovo-Mumford regularity and d-normality. This yields a characterization of the singular locus of X-2,X-d,X-n and X-3,X-2,X-n