Terracini loci of tangential varieties of Veroneseans

Abstract

The Terracini locus of a projective variety tells us the sets of points at which Terracini's lemma will fail to recover the dimension of the corresponding secant variety. We study the Terracini loci of tangential varieties of Veronese varieties, classifying several cases, including a complete classification of the Terracini loci of tangential varieties to rational normal curves. We also show that many Terracini loci involve only sets with "intratangent pairs" that lie on the same tangent space to the underlying Veronese variety. Finally, we give some applications to algebraic statistics, showing that osculating varieties of Veronese varieties are moment varieties for local mixtures of location families and connecting Terracini loci to local identifiability.