On the \(X\)-rank with respect to linearly normal curves (Q1956279)

The authors study the rank of points of a projective space, with respect to a curve \(X\subset\mathbb P^n\). The rank of a point \(P\) is defined as the minimal integer \(r\) for which there are points \(P_1,\dots, P_r\in X\) with \(P=P_1+\dots+P_r\). When \(X\) is a variety that parametrizes simple objects (like decomposable tensors or powers of linear forms), then the rank provides a measure of the complexity of \(P\). The authors consider linearly normal irreducible curves \(X\) and study bounds for the rank with respect to \(X\). They prove that if \(n\geq 2s+2\) and the intersection of \(X\) with the span \(L=\langle P_1,\dots, P_s\rangle\) has degree at most \( \deg(X)-2p_a(X)\), then the rank of a general point in \(L\) is bounded by \(n+1-s\). The authors also study in details the case of smooth curves \(X\) of genus \(2\) and degree \(n+2\) (\(n\geq 8\)). They prove that, for such curves, the rank of points in the tangential variety \(T_X\) (and not in \(X\)) is exactly equal to \(n-2\).