Terracini loci and homogeneous spaces Terracini Loci et espaces homogènes

We study the linear dependence of disjoint unions of double points of an integral and non-degenerate variety X ⊂ Pr. Such sets are called Terracini loci. Our main results are for Segre-Veronese embeddings and a few other homogeneous spaces. To study the minimal number of such double points which are linearly dependent, it is useful to study the minimal degree curves contained in X. We give an example (the Segre embedding of P1 × P1) in which these curves are not sufficient to describe these Terracini loci.