On the Terracini locus of a Segre variety

The k-th Terracini locus of a variety X measures the degeneracy of the canonical map between the abstract k-th secant variety and the k-th secant variety of X.

After introducing this locus, we focus on the case of Segre varieties and we determine the second and third Terracini locus of any Segre variety. Lastly, we explain how a better knowledge of the Terracini locus is essential when dealing with algorithms computing tensor rank decomposition.