Ideals of varieties parameterized by certain symmetric tensors

Overview & Synopsis

The ideal of a Segre variety $\PP {n_{1}}\times \cdots \times \PP {n_{t}}\hookrightarrow \PP {(n_{1}+1)\cdots (n_{t}+1)-1}$ is generated by the $2$-minors of a generic hypermatrix of indeterminates (see \cite{Ha1} and \cite{Gr}). 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 $2$.