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 ⊂ 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.

A footnote to a footnote to a paper of B. Segre

“…, L 9 }. If we identify each L i with a linear form in R, then there exists a linear combination F = a 1 L 6 1 + • • • + a 9 L 6 9 and we know that the linear combination is unique, for v 6 (Z) is linearly independent. Compute the a i 's.…”

“…The homogeneous ideal of the residue B of A in the intersection is obtained by erasing the bottom row of M and adding a column of three quadrics. The maximal minors of the matrix M ′ that we obtain generate the homogeneous ideal of a scheme B linked to A in a complete intersection (3,6).…”

“…Since by b3) Dh Z (6) > Dh Z (7) ≥ 1, then the h-vector of Z is (1, 2, 3, 3, 3, 3, 2, 1). By [3] Lemma 5.3 we know that Z has the Cayley-Bacharach property CB (6). Since moreover the h-vector of Z is the same than the h-vector of a complete intersection of type (3,6), by [16] Z is complete intersection of a cubic and a sextic curve.…”

“…By [3] Lemma 5.3 we know that Z has the Cayley-Bacharach property CB (6). Since moreover the h-vector of Z is the same than the h-vector of a complete intersection of type (3,6), by [16] Z is complete intersection of a cubic and a sextic curve. Then A and B are linked by a cubic and a sextic curve.…”

“…The phenomenon can be investigated in terms of the map s 9 from the abstract secant variety AS 9 (X) to S 9 (X), which is generically 2:1. The component of the ramification locus R is strictly connected with the theory of Terracini loci, introduced in [6]. Roughly speaking, Terracini loci contain finite subsets of X such that the tangent spaces to the points are not independent.…”