Two Denegeration Techniques for Maps of Curves

In this paper, we discuss the theories of admissible covers (HarrisMumford) and limit linear series (Eisenbud-Harris), two techniques for using degenerations to study maps between curves. While the theories are superficially very similar, they have certain fundamental differences which we will also explore. 1. Motivational problems We begin by laying out a number of questions on curves which should serve as motivation for the degeneration theories we discuss. Not all the problems mentioned require the use of admissible covers or limit linear series, or even degeneration arguments in general. However, they may all be approached by these methods, typically producing very natural solutions. One fundamental question is the following: Question 1.1. What kinds of maps exist from curves to other curves? To projective spaces? The first part of the question leads naturally to the theory of branched covers of curves, while the second leads to the theory of linear series. In the case of maps to the projective line, both theories apply and the interplay between them is often fruitful, particularly in positive characteristic, where even basic questions remain open [12], [11], [9]. The fundamental distinction in this context is that the theory of branched covers treats prescribed branching on the target, while the theory of linear series deals with ramification on the source. However, we will see that the techniques and results of both have a number of important distinctions. First, we discuss some of the more specific directions in which one can take Question 1.1. One could make the question more precise by, for instance, fixing the target curve and the branch behavior of a map, and asking if there is a source curve giving such a map; this is closely tied to the theory of fundamental groups [1]. Or, one could fix the dimension of the projective space and the degree of a prospective map 2000 Mathematics Subject Classification. 14H30.