The effective cone on symmetric powers of curves

Abstract

The dth symmetric power C_d of a smooth complex projective curve (or compact Riemann surface) C is a parameter space for effective divisors of degree d on C, so that the theory of degree-d maps from C to projective space is encoded in the subvarieties of C_d and the relations amongst them. We give a complete description of the cone of codimension-1 subvarieties of C_g–1 when C is a general curve of genus g ≥ 4, as well as new bounds for the case C_d in the range ([special characters omitted], g – 2]. We also give new information on the movable cone of C_d and the volume function of C_g–1.