Singular vector tuples and their geometry

The main topic of this thesis is the geometry of singular vector tuples of tensors. Singular vector tuples are a generalization of singular pairs of matrices to higher-order tensors. The singular vector tuples of a tensor T consist of the critical points of the function d(T, X) that measures the distance between the tensor T and the Segre-Veronese variety X, where the distance is the one defined by the Bombieri-Weyl product on the tensor space. The main question of this work is to answer the question: are tensors determined by their singular vector tuples? For partially symmetric tensors the answer is positive if some degree is odd. On the other hand, if all degrees are even there exists a one-dimensional family of tensors with the same singular vector tuples. Another important fact of the geometry of singular vector tuples is that the tensor T itself is a linear combination of them when the dual variety of the Segre-Veronese variety is non-defective, however, when such condition is disregarded the answer is not known. Utilizing cohomological techniques, together with symbolical computation in Macaulay2, we show that this property remains true in the first examples where the dual of the Segre-Veronese variety is defective.