Communications of the
Korean Mathematical Society CKMS

The Terracini $t$-locus of an embedded variety $X\subset \mathbb{P}^r$ is the set of all cardinality $t$ subsets of the smooth part of $X$ at which a certain differential drops rank, i.e., the union of the associated double points is linearly dependent. We give an easy to check criterion to exclude some sets from the Terracini loci. This criterion applies to tensors and partially symmetric tensors. We discuss the non-existence of codimension $1$ Terracini $t$-loci when $t$ is the generic $X$-rank.

Keywords: Partially symmetric tensor, Terracini locus, secant variety, Segre variety, multiprojective space

MSC numbers: Primary 14N05, 14N07, 15A69