The Segre variety \(X = \mathrm{Seg}({\mathbb P}V^*_1 \times \dots \times {\mathbb P}V^*_n)\) is the image of the Segre embedding of a product of projective spaces of dimension \(r_i\) into \({\mathbb P}^N\), \(N=\prod_i^n (r_i+1) -1\); \(X\) parameterizes decomposable \((r_1+1)\times \dots \times (r_n+1)\)-tensors, i.e. rank 1 tensors.
In this paper the tangential variety \(\tau (X)\) is studied (the union of all projective tangent lines to \(X\)). A conjecture by Landsberg and Weyman describes how the ideal of \(\tau (X)\) should be generated, and the main result here is to prove a set theoretic version of such conjecture.
Namely, it is shown that \(\tau (X)\) is set theoretically the intersection of hypersurfaces of degrees 2,3 and 4, where the quadric equations are linear combinations of \(2\times 2\) minors of a \((r_1+1)\times \dots\times (r_n+1)\)-tensor of indeterminates \(T\), the cubic ones are linear combination of \(3\times 3\) minors of a flattening of \(T\) and the quartic ones comes from hyperdeterminants of format \(2\times 2 \times 2\) in \(T\). More precisely, the equations come from the linear span of the \((SL(2)^{\times n})\times \mathfrak S_n\)-orbits respectively of a quadratic form \(F_0\), a cubic \(F_2\) and a Cayley \(2\times 2 \times 2\)-hyperdeterminant, where \(F_0\) is a linear combination of \(2\times 2\) minors and \(F_2\) is a linear combination of \(3\times 3\) minors of a flattening.
One of the main ingredients for the proof is the study of the variety of principal minors of \(n\times n\) symmetric matrices which contains \(\tau (X)\) and is defined by the quartic equations above.