On the 𝑋-rank of a curve 𝑋⊂ℙⁿ: an extremal case

Abstract

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