Journal of the
Korean Mathematical Society JKMS

We offer a refinement of the classical Clifford inequality about special linear series on smooth irreducible complex curves. Namely, we prove about curves of genus $g$ and odd gonality at least 5 that for any linear series $g^r_d$ with $d \leq g+1$, the inequality $3r \leq d$ holds, except in a few sporadic cases. Further, we show that the dimension of the set of curves in the moduli space for which there exists a linear series $g^r_d$ with $d < 3r$ for $d \leq g+l$, $0 \leq l \leq \frac{g}{2}-3$, is bounded by $2g-1 + \frac{1}{3} (g+2l+1)$.

Keywords: gonality, divisors, Clifford inequality, special linear series

MSC numbers: Primary 14H51; Secondary 14C20