Terracini Loci for Maps

Abstract

Let X be a smooth projective variety and f:X→Pr a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case dimX=1. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set B⊊X or solutions containing a prescribed p∈X). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools.

Keywords

• Terracini loci
• curve
• curves in projective spaces
• joins of varieties
• secant varieties