On a theorem of Castelnuovo and applications to moduli

TL;DR

This paper proves a classical theorem bounding the dimension of plane curve linear systems based on genus and other invariants, classifies systems within certain dimension ranges, and applies results to determine the maximum moduli of curves in linear systems on surfaces.

Contribution

It provides a proof of Castelnuovo's theorem, classifies linear systems with specific dimensions, and solves a moduli problem for curves on surfaces, especially for genus g ≥ 22.

Findings

Bound on dimension of linear systems in terms of genus.

Classification of linear systems with certain dimensions.

Maximum moduli of curves in linear systems is 2g+1 for g ≥ 22.

Abstract

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. Then we classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo's theorem. Finally we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.