On the strong base locus of a projective variety

TL;DR

This paper introduces and analyzes the concepts of base locus and strong base locus of a projective variety, exploring their properties, connections to tangential projections, and applications to tensor-related varieties like Veronese and Segre-Veronese.

Contribution

It defines the notions of base locus and strong base locus, studies their properties, and characterizes their nonemptiness for specific tensor-related varieties, advancing understanding of tangential contact and interpolation.

Findings

Defined base locus and strong base locus for projective varieties.

Connected these notions to tangential projections and interpolation problems.

Characterized nonemptiness of loci for Veronese and Segre-Veronese varieties.

Abstract

We introduce and study the base locus and the strong base locus of a projective variety X. The base locus of X parametrizes configurations of smooth points of X where the span of the tangent spaces of X at these points intersects X at some additional smooth point. The strong base locus parametrizes configurations of smooth points of X for which the span of the tangent spaces of X at the given configuration contains the entire tangent space at an additional point. These notions originate from the study of base loci of tangential projections, are strictly related to interpolation problems with double points in special position, and provide a natural framework to study tangential contact for nongeneral points. We give first properties and explore connections with Terracini loci and with the concept of identifiability. We focus on tensor-related varieties and characterize the nonemptiness of base loci and strong base loci for Veronese and Segre-Veronese varieties.