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Take a smooth, connected and non-degenerate projective curve $X\subset\mathbb {P}^r$, $r\ge 2b+2\ge 6$, defined over an algebraically closed fieldwith characteristic $0$ and let $\sigma _b(X)$ be the $b$-secant variety of$X$. We prove that the $X$-rank of $q$ is at least $b+1$ for a non-emptycodimension $1$ locally closed subset of $\sigma _b(X)$.

The $b$-secant variety of a smooth curve has a codimension $1$ locally closed subset whose points have rank at least $b+1$