Appendix to “Secant varieties and successive minima" by C. Soulé: On linear subspaces contained in the secant varieties of a projective curve

We show that the $d$-th secant variety of a projective curve of genus $g$ imbedded in projective space by a complete linear system of degree $2g-2+m$, with $m$ at least $2d+3$, does not contain linear spaces of dimension bigger than $d-1$, and that the only linear subspaces of dimension $d-1$ contained in it are the obvious ones.