On tensors that are determined by their singular tuples

In this paper we study the locus of singular tuples of a complex-valued multisymmetric tensor. The main problem that we focus on is as follows: Given the set of singular tuples of some general tensor, what are all the tensors that admit those same singular tuples? Assume that the triangle inequality holds, which is exactly the condition that the dual variety to the Segre--Veronese variety is a hypersurface, or equivalently, the hyperdeterminant exists. We show in such cases that, when at least one component has degree odd, this tensor is projectively unique. On the other hand, if all the degrees are even, the fiber is a 1-dimensional space.