Vector Bundles with Sections

Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r + 1 global sections on a curve of genus g. After reviewing the main results in this theory, and the role of degeneration techniques in proving them, we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open. Focusing on the case of rank 2, we will discuss the role of spaces with fixed determinant, and how the dimension theory of Artin stacks may be useful in applying degeneration techniques. 1. Classical Brill-Noether theory Classical Brill-Noether theory studies linear series on a general smooth projective curve C of genus g. We recall: Definition 1.1. A linear series of dimension r and degree d on C is a pair (L, V ) where L is a line bundle of degree d on C, and V ⊆ H0(C,L) is an (r+ 1)-dimensional space of global sections. Gd(C) is the space of all linear series of dimension r and degree d on C; it is a projective scheme over Pic(C). Gd(C) is a compactification (in the weak sense) of the space of non-degenerate morphisms C → Pr of degree d. Beyond the immediate applications to such morphisms, Eisenbud and Harris successfully applied linear series techniques to topics such as the existence of curves with particular types of Weierstrass points (which can be rephrased in terms of the geometry of the canonical imbedding), and the geometry of the moduli space of curves (in particular, that it is of general type for g ≥ 24). The space Gd(C) can be constructed inside a relative Grassmannian scheme over Pic (C), which shows that the dimension of any component of Gd(C) is at least ρ = (r + 1)(d− r)− rg. The foundational results of Brill-Noether theory can be summarized as follows. Theorem 1.2. Let C be a smooth projective curve of genus g, and fix d, r > 0. Then: (1) Gd(C) is non-empty if ρ ≥ 0; (2) Gd(C) is connected if ρ ≥ 1; (3) Gd(C) is pure of dimension ρ if C is general; (4) Gd(C) is smooth if C is general. Note that (3) implies in particular that Gd(C) is empty if ρ < 0 and C is general. These statements were proved over the course of a decade from the early 1970’s through the early 1980’s. Many of them go back to Brill and Noether in 1874, and the original arguments follow ideas due to Castelnuovo and Severi. Kempf and Kleiman-Laksov settled (1) in 1971; Griffiths and Harris, building on further work of Kleiman, proved (3) in 1980, Fulton and Lazarsfeld proved (2) in 1981, and Gieseker proved (4) (reproving (3) in the process) in 1982 by proving an assertion of Petri dating back to 1925. Assertions (1) and (3) together are frequently called the “Brill-Noether theorem.”