  We apply a degenerate version of a result due to Hirschowitz, Ramanan and
Voisin to verify Green and Green-Lazarsfeld conjectures over explicit open sets
inside each $d$-gonal stratum of curves $X$ with $d<[g_X/2]+2$. By the same
method, we verify the Green-Lazarsfeld conjecture for any curve of odd genus
and maximal gonality. The proof invokes Voisin's solution to the generic Green
conjecture as a key argument.
