  The present paper is related to a conjecture made by Green and Lazarsfeld
concerning 1-linear syzygies of curves embedded by complete linear systems of
sufficiently large degrees. Given a smooth, irreducible, complex, projective
curve $X$, we prove that the least integer $q$ for which the property $(M_q)$
fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree
becomes sufficiently large. Consequently, this number is an invariant of the
curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to
saying that this invariant equals the gonality of the curve. We verify the
conjecture for plane curves, curves lying on Hirzebruch surfaces, and for
generic curves having the genus sufficiently large compared to the gonality. We
conclude the paper by proving that Green's canonical conjecture holds for
curves lying on Hirzebruch surfaces.
