A Segre-Veronese variety \(X\) is the embedding of a product of projective spaces \(\mathbb{P}(V_1)^*\times \dots\times \mathbb{P}(V_n)^*\) into \(\mathbb{P}^N\), via the line bundle \(\mathcal{O}(d_1,\dots,d_n)\), where \(N=\prod_{i=1}^n{m_i+d_i\choose m_i} -1\) and \(m_i+1=\dim V_i\). In particular, if \(n=1\), then \(X\) is a Veronese variety, given by the \(d_1\)-ple embedding of \(\mathbb{P}^{m_1}\), while if \((d_1,\dots,d_n) = (1,\dots,1)\), \(X\) is a Segre variety. In the first case, \(X\) parameterizes symmetric \(d\)-dimensional \((m_1+1)\times \dots\times (m_1+1)\)-tensors of symmetric rank 1 (equivalently, degree \(d\) forms in \(m_1+1\) variables which are \(d_i\)-powers of a linear form); in the second, a Segre variety \(X\) parameterizes \((m_1+1)\times \dots\times (m_n+1)\) tensors of tensor rank 1. In the general case, a Segre-Veronese variety \(X\) parameterizes partially symmetric filtrations of rank 1. It us known that the ideal of such varieties are generated by \(2\times 2\) minors of a generic tensors with given symmetries; it was conjectured by Garcia, Stillman and Sturmfelds that the ideal of the secant line variety \(\sigma_2(X)\), in the Segre case, is generated by the \(3\times 3\) minors of “flattenings” of a generic tensor of indeterminates (roughly speaking, one slices the tensor into 2-dimensional slices and then lines up the slices to form an ordinary matrix and considers its \(3\times 3\) minors). For Veronese varieties, the fact that this procedure gives the ideal of \(\sigma_2(X)\) has been proved by Kanev, while the set-theoretic version of the conjecture has been proved by Landsberg and Manivel. In this paper the conjecture is proved to hold not only for Segre-Varieties, but also for Segre-Veronese ones.
The proof uses combinatorics and representation theory; it is mainly based on the possibility of obtaining the decomposition of the coordinate ring of \(\sigma_2(X)\) into irreducible representations (under the action of a product of generic linear groups), a result that also allows to compute the Hilbert function of \(\sigma_2(X)\).