Let \( X \) be the Segre-Veronese embedding of \(\mathbb{P}^a\times\mathbb{P}^b\times\mathbb{P}^c\times \cdots\) with degree \( L=(l_1,l_2,l_3,\dots).\) Corresponding to it we have the multigraded ring \[ S[X]=\mathbb{C}[\alpha_0,\dots,\alpha_a,\beta_0,\dots,\beta_b,\gamma_0,\dots,\gamma_c,\dots] .\]
The ring \( S[X] \) has two dual interpretations. The first, more geometric, as “functions” on \(X\) and the second, more algebraic, in terms of derivations. We have then the well known apolarity lemma: \[F\in \langle \{ p_1,\dots,p_r \} \rangle \Leftrightarrow I(\{ p_1,\dots,p_r \}) \subset \mbox{Ann}(F)\] where \(F\in S[X], \ p_1,\dots,p_1\in X\) and \(\mbox{Ann}(F)\subset S[X]\) is the annihilator of \(F\) in the space of derivations.
This lemma gives a characterization of rank \(r\) tensors. In the present paper the authors introduce a result for border rank similar to the apolarity lemma. The main result goes as follows: Suppose a tensor or polynomial \(F\) has border rank at most \(r.\) Then there exists a multihomogeneous ideal \(I\subset S[X]\) such that \(I\subset \mbox{Ann}(F)\) and for each multidegree \(D\) the \(D\)th graded piece \(I_D\) of \(I\) has codimension equal to \(\min(r,\dim S[X]_D).\)
In fact, the result stated above is a consequence of more general results that they show for a smooth toric variety. They also have if and only if kind of results using more technical hypotheses.
Moreover, the results are applied to the cases \(\mathbb{P}^2\times \mathbb{P}^1 \times \cdots \times \mathbb{P}^1\) with arbitrary degree and \(\mathbb{P}^a\times \mathbb{P}^b\times \mathbb{P}^c\) with degree \((1,1,1)\) obtaining new bounds for the border rank of such Segre-Veronese varieties.