Luca Sodomaco: Relative tangency and Euclidean distance data loci

Abstract.

In this talk, we introduce a relative notion of tangency of a projective variety X with respect to a subvariety Z not contained in the singular locus of X. In particular, we define the notion of projective dual variety of X relative to Z, and we introduce the polar classes of X relative to Z. We concentrate on the duality of determinantal varieties relative to special linear sections.

The main application of this toolkit is the study of critical points of the Euclidean Distance (ED) function from a data point to an assigned real algebraic variety X. In particular, we focus on loci of data points admitting at least one ED critical point on a prescribed subvariety of X. These are called ED data loci of affine and projective varieties. We show their irreducibility, and we compute their dimensions and degrees.

This talk is based on an upcoming work joint with Lukas Gustafsson and Sandra Di Rocco.