On the Terracini Locus of Projective Varieties

Abstract

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite sets $$S \subset X$$ S ⊂ 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.

Terracini Loci and Homogeneous Spaces

“…See [3,13] for the use of arrows for a tangent space computation and hence an improvement of [18]. Recently, inspired by Terracini's lemma there was a study of the Terracini locus T 1 (X, x) ( [4,5]). The main motivation came from the observation that it is easy to compute tangent spaces using a computer and that these computations help to check the identifiability of a tensor T (resp.…”

Decompositions and Terracini loci of cubic forms of low rank

“…, 2x r } do not impose independent conditions on O X (1). In fact, with this formulation, one can give the definition of Terracini locus for an abstract variety and any (ample) line bundle, see [BC21].…”

A note on very ample Terracini loci

“…Terracini loci were introduced by the first author and Chiantini in [2]. Their emptiness implies non-defectivity of secant varieties due to the celebrated Terracini's lemma, whereas the converse is not true: there exist non-empty Terracini loci even in the presence of non-defective secants.…”