The flex divisor of a K3 surface

The flex divisor of a primitively polarized K3 surface $(X,L)$ of degree $L^2=2d$ is, generically, the locus of all points $x\in X$ for which there exists a pencil $V\subset |L|$ whose base locus is $\{x\}$. We show that the flex divisor lies in the linear system $|n_dL|$ where $n_d=(2d+1)C(d)^2$ and $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $F_{2d}$ of polarized K3 surfaces.