Genus bound for low-degree curves on complete intersection threefolds

Let $X=Z(f_1,\dots,f_{n-3})\subset\mathbb{P}^n$ be a complete intersection threefold, where $f_i$ has degree $k_i$, and let $C\subset X$ be a curve of degree $d$ and genus $g$. The genus-bound conjecture. If

$$d\leq\frac{1}{2}(k_1\cdots k_{n-3}),$$

then

$$g\leq\frac{d}{2}(k_1+\cdots+k_{n-3}-n-1)+\frac{2d}{3}+1.$$

This is presented as a special case of the Bayer–Macrì–Toda inequality for the ideal sheaf of $C$. The paper notes that the bound is expected from Bridgeland stability and that a broader genus bound for curves of arbitrary degree is conjectured subsequently.