Symmetric tensor rank with a tangent vector: a generic uniqueness theorem (Q2845855)

Let \(f\in k[x_0,\dots,x_m]_d\) be a generic element which can be written as \(f=\sum_{i=1}^{t-2}L_i + L_{t-1}^{d-1}L_t\), where the \(L_i\)'s are linear forms. For \(m\geq2\), \(d\geq 7\) and certain values of \(t\) (\(3\leq t\leq \lfloor {m+d-2\choose m}/(m+1)\rfloor \)), it is proved in this paper that the linear forms \(L_i\) are uniquely determined (up to scalar multiplications).NEWLINENEWLINEThis can be expressed in a more geometric way, as follows: Let \(X_{m,d}\subset \mathbb{P}^N\), \(N={m+d\choose m}-1\), be the \(d\)-uple Veronese embedding of \(\mathbb{P}^m\) (\(X_{m,d}\) parameterizes the forms \(f\) of degree \(d\) of type \(f=L^d\), for \(L\in k[x_0,\dots,x_m]_1\)) and let \(\tau(X_{m,d})\) be its tangential developable. Consider the variety \(\tau(X_{m,d},t)\) which is the join of \(\tau(X_{m,d})\) and \(t-2\) copies of \(X_{m,d}\); then the previous result amounts to saying that given a generic \(P\in \tau(X_{m,d},t)\), there exist and are uniquely determined (for \(d,m,t\) satisfying the bounds above) points \(P_1,\dots,P_{t-2}\in X_{m,d}\) and a tangent line \(l\) to \(X_{m,d}\), such that \(P\in <P_1,\dots,P_{t-2},l>\).NEWLINENEWLINEThe proof is based on the possibility of transposing these geometric properties into statements about subschemes of \(\mathbb{P}^m\) and their Hilbert function.