On the Terracini locus of projective varieties

TL;DR

This paper investigates the properties and structure of the Terracini locus of projective varieties, focusing on criteria for membership, bounds on dimension, and specific examples for Veronese varieties.

Contribution

It introduces new criteria for identifying elements in the Terracini locus and provides bounds and explicit examples for Veronese varieties.

Findings

Criteria to exclude sets from the Terracini locus

Bounds on the dimension of the Terracini locus for Veronese varieties

Examples where the locus has codimension 1

Abstract

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite subsets S of X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties. We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.