Terracini Loci of Segre Varieties

Abstract

Fix a format (n1+1)×⋯×(nk+1), k>1, for real or complex tensors and the associated multiprojective space Y. Let V be the vector space of all tensors of the prescribed format. Let S(Y,x) denote the set of all subsets of Y with cardinality x. Elements of S(Y,x) are associated to rank 1 decompositions of tensors T∈V. We study the dimension δ(2S,Y) of the kernel at S of the differential of the associated algebraic map S(Y,x)→PV. The set T1(Y,x) of all S∈S(Y,x) such that δ(2S,Y)>0 is the largest and less interesting x-Terracini locus for tensors T∈V. Moreover, we consider the one (minimally Terracini) such that δ(2A,Y)=0 for all A⊈S. We define and study two different types of subsets of T1(Y,x) (primitive Terracini and solution sets). A previous work (Ballico, Bernardi, and Santarsiero) provided a complete classification for the cases x=2,3. We consider the case x=4 and several extremal cases for arbitrary x.