Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.

Consider the variety $S_{3}\subset\mathbb{G}(1,3)$, in the Grassmannian of lines in $\mathbb{P}^{3}$, parametrizing lines which are $3$-secant to $C$. Then $dim(S_3) = 1$.

Does there exists a formula involving $d$ and $g$ for the geometric genus $g(S_{3})$ ?