On the nef cone of symmetric products of a generic curve

The nef cone of the symmetric product C^(k) of a curve C with general moduli is easily determined when k is at least equal to the gonality gon(C) of the curve. In this paper we describe the nef cone for k = gon(C) - 1, when the genus of C is even. In this case the boundary of the nef cone has rational slope and is determined using curves on C^(k) associated to the g¹_k+1 's of C.