Limit Linear Series: Constructions and Applications

We discuss the Eisenbud-Harris theory of limit linear series, and survey a number of applications, ranging from the geometry of moduli spaces of curves to connectedness of certain Hurwitz spaces. We also present a more recent construction of limit linear series, and discuss applications and potential applications both to classical linear series and to their higher-rank generalization. In this series of lectures, we will survey the theory of linear series on curves, with a particular emphasis on degeneration techniques. We will also discuss the generalization to higher-rank vector bundles. The lecture plan is as follows: (1) Review of the basics of linear series on smooth curves and their ramification; discusssion of the theory of limit linear series developed by Eisenbud and Harris. (2) Discussion of several different applications of the Eisenbud-Harris theory. (3) Presentation of a more recent approach to limit linear series, and examination of the relationship to the Eisenbud-Harris approach. (4) Construction of linear series and limit linear series spaces, and the related notion of linked Grassmannians. (5) An application of the new construction to fibers of generalized Abel maps; discussion of higher-rank Brill-Noether theory, and approaches to it via degeneration techniques. We work throughout over an algebraically closed field F of characteristic 0, although we will periodically remark on the situation for positive characteristic, and some of the foundational results hold over much more general schemes. 1. Eisenbud-Harris limit linear series In this lecture, we begin by recalling basic definitions relating to linear series on smooth curves and ramification sequences, and preview some of the applications of the theory. We then discuss the theory of limit linear series developed by Eisenbud and Harris, which describes how linear series specialize under degenerations from smooth curves to curves of compact type. 1.1. Linear series on curves. Linear series arise naturally from the study of maps of varieties into projective space. Specifically, if X is a smooth proper curve over F , then F -linear maps from X to Pr of degree d correspond to pairs (L , {v0, . . . , vr}) of line bundles L on X of degree d together with sections {v0, . . . , vr} ∈ H0(X,L ) such that for every P ∈ X, at least one of the vi is nonvanishing at P . The map is non-degenerate (that is, does not have image contained in a hyperplane) if and only if the vi are linearly independent, and if we wish to work up to automorphism of Pr, we replace the vi by the (r + 1)-dimensional vector space V which they span. Finally, it turns out we can obtain a natural compactification of this space simply by dropping the requirement that V be non-vanishing at every point of X. Thus motivated, we have: Definition 1.1.1. A linear series of degree d and dimension r (also called a gd) on X is a pair (L , V ) where L is a line bundle of degree d on X, and V is an (r + 1)-dimensional subspace of H0(X,L ). We also briefly mention the idea of ramification, which plays an important role later: