Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety Pn1×⋯×Pnt↪P(n1+1)⋯(nt+1)−1Pn1×⋯×Pnt↪P(n1+1)⋯(nt+1)−1 is generated by the 2-minors of a generic hypermatrix of indeterminates (see [H.T. Hà, Box-shaped matrices and the defining ideal of certain blowup surface, J. Pure Appl. Algebra 167 (2–3) (2002) 203–224. MR1874542 (2002h:13020)] and [R. Grone, Decomposable tensors as a quadratic variety, Proc. Amer. Math. 43 (2) (1977) 227–230. MR0472853 (57 #12542)]). We extend this result to the case of Segre–Veronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veronese varieties from a Cohen–Macaulay subvariety of codimension 2.