Irreducibility and monodromy of some families of linear series

Abstract

— Let g, r, and d be positive integers such that g=(r-{-1) (g—d-\-r), so that the general curve of genus g has only finitely many g^s. We will show in this paper that for suitable families of curves ^ -> B, the family of all cffs on all fibers of ^ -> B is irreducible. We do this by analyzing the monodromy action on the set of 97 on a fibre, using a degeneration to reducible curves and our technique of limit series [198?^]. In the case r = 1 we prove the sharper statement that the monodromy is the full symmetric group, a result motivated by a problem posed by Verdier, and applied by him in the study of harmonic maps from 2 to S (Verdier [198?]). If we take ^ to be the universal curve over a suitable open set B of the moduli space My then the family of c^'s is a finite cover of B, and the branch locus of this cover (in the case r=l), analyzed through the tools developed in this paper, plays a fundamental role in the even-genus case in our proof [198?^] that Jig has general type for all g 24.