Clifford index of smooth algebraic curves of odd gonality with big $W^{r}_{d}(C)^*$

Abstract

(Received July 19, 2000)0. IntroductionLet be a smooth projective irreducible algebraic curve over the eld of com-plex numbers C or a compact Riemann surface of genus . Let ( ) be the Jacobianvariety of the curve , which is a -dimensional abelian variet y parameterizing all theline bundles of given degree on . We denote by ( ) a subvariety o f the Jaco-bian variety ( ) consisting of line bundles of degree with +1 o r more independentglobal sections.If > + 2, one can compute the dimension of ( ) by using the Riemann-Roch formula, and this dimension is independent of . If  + 2, the di-mension of ( ) is known to be greater than or equal to the Brill- Noether number( ) := ( + 1)( + ) for any curve , and is equal to ( ) forgeneral curve by theorems of Kleiman-Laksov [13] and Grift hs-Harris [7]. On theother hand, the maximal possible dimension of ( ) for this ran ge of , and is 2 and the maximum is attained if and only if is hyperelliptic b y a well knowntheorem of H. Martens [16].From a result of M. Coppens, G. Martens and C. Keem [4, Corollary 3.3.2], it isknown that for curves of odd gonality — i.e. curves for which t he minimal numberof sheets of a covering over P