On the $X$-rank of a curve $X\subset \mathbb {P}^n$: an extremal case

Abstract:

Let $X\subset \mathbb {P}^n$, $n \ge 3$, be an integral and non-degenerate curve. For any $P\in \mathbb {P}^n$ the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset Y$ such that $P$ is in the linear span of $S$. Landsberg and Teitler proved that $r_X(P) \le n$ for any $X$ and any $P$. Here we classify the pairs $(X,Q)$, $Q\in X_{reg}$, such that all points of the tangent line $T_QX$ (except $Q$) have $X$-rank $n$: $X \cong \mathbb {P}^1$ and $T_QX$ has order of contact $\deg (X) +2-n$ with $X$ at $Q$.