Terracini Loci and Homogeneous Spaces

We study the linear dependence of disjoint unions of double points of an integral and non-degenerate variety $$$X\subset ℙ^r$$$. 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$$$\times$$$ ℙ¹) in which these curves are not suffcient to describe these Terracini loci.

We study the linear dependence of disjoint unions of double points of an integral and non-degenerate variety $$$X\subset ℙ^r$$$. 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$$$\times$$$ ℙ¹) in which these curves are not suffcient to describe these Terracini loci.

Terracini’s lemma homogeneous spaces secant varieties zerodimensional schemes tangent spaces

Terracini’s lemma homogeneous spaces secant varieties zerodimensional schemes tangent spaces