On a theorem of Castelnuovo and applications to moduli

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. This extends a previous result of Castelnuovo and Enriques.We classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo’s theorem. Then we make an application to the followingmoduli problem: what is themaximu mnumber ofmoduli of curves of geometric genus g varying in a linear system on a surface? It turns out that, for g ≥ 22, theanswer is 2g+1, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.