Multisecant subspaces to smooth projective varieties in arbitrary characteristic

Let $X \subseteq \mathbb{P}^{N}$ be a projective variety of dimension $n\geq 1$, degree $d$, and codimension $e$, not contained in any hyperplane, defined over an algebraically closed field $\Bbbk$ of arbitrary characteristic. We show that if a $k$-dimensional linear subspace $M$ meets $X$ at the smooth locus such that $X\cap M$ is finite and locally lies on a smooth curve, then the length $l(X\cap M)$ does not exceed $d-e+k-\min \{g,e-k\}$ for the sectional genus $g$ of $X$.