Green–Lazarsfeld gonality conjecture

Let $C$ be a curve of gonality $d$, and let $L$ be a nonspecial very ample line bundle on $C$. Writing $h^0(L)$ for the dimension of its space of global sections, Green–Lazarsfeld's gonality conjecture. There exists such an $L$ for which

$$K_{h^0(L)-d,1}(C,L)=0.$$

This predicts optimal syzygy vanishing for a sufficiently positive embedding of a curve, with the gonality determining the last nonvanishing linear syzygy. The source presents it as a conjecture for bundles of large degree.