Virtual commutative algebra semnar
27 January 2023, 5:30 pm
For getting the Google meet link
Please contact the organisers and register by filling a form at the site
https://sites.google.com/view/virtual-comm-algebra-seminar/home
Host: J. K. Verma
Speaker: Alessio Caminata
Affiliation: University of Genoa, Italy
Title: 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.