Determinantal varieties from point configurations on hypersurfaces

Abstract: Point configurations appear naturally in different contexts, ranging from the study of the geometry of data sets to questions in commutative algebra and algebraic geometry concerning determinantal varieties and invariant theory. In this talk, we bring these perspectives together: we consider the scheme X_{r,d,n} parametrizing n ordered points in r-dimensional projective space that lie on a common hypersurface of degree d. We show that this scheme has a determinantal structure and, if r>1, 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 complete characterization of the singular locus of X_{2,d,n} and X_{3,2,n}. This is joint work with Han-Bom Moon and Luca Schaffler.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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