The gonality of general smooth curves with a prescribed plane nodal model

Abstract

Let C be a smooth curve of genus g. It is well-known that C possesses a linear system g l if Q l(g) = 2n g 2 ~ 0. Moreover, if C is general and Q~(g) 2 ( k + l ) and 6<kd(k+l )2+3 (Theorem 2.1 in loc. cit.) and for each d this bound on 6 is sharp (Examples 4.1 and 4.2 in loc. cit.). In this paper we consider the question for a general choice of F. Since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible (famous conjecture of Severi proved by Harris [9]) the meaning of general becomes "represented by a point of a particular nonempty Zariski-open subset of that parameter-space". Note that Qn x_ 3(g) < 0 if and only if 6 < (d 2 7d + 18)/2.