A REFINEMENT OF THE CLASSICAL CLIFFORD INEQUALITY

Abstract

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 <TEX>$g^r_d$</TEX> with <TEX>$d{\leq}g+1$</TEX>, the inequality <TEX>$3r{\leq}d$</TEX> 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 <TEX>$g^r_d$</TEX> with d<3r for <TEX>$d{\leq}g+l,\;0{\leq}l{\leq}\frac{g}{2}-3$</TEX>, is bounded by <TEX>$2g-1+\frac{1}{3}(g+2l+1)$</TEX>.