Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety $\mathbb{P}^{n_{1}}\times \cdots \times\mathbb{P}^{n_{t}}\hookrightarrow \mathbb{P}^{(n_{1}+1)\cdots (n_{t}+1)-1}$ is generated by the $2$-minors of a generic hypermatrix of indeterminates. 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 $2$.