Vector Bundles and Brill–Noether Theory

After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose several problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical determinant with five linearly independent global sections on a non-tetragonal curve of genus 7 is a smooth Fano threefold of genus 7. As a natural generalization of line bundles, vector bundles have two important roles in algebraic geometry. One is the moduli space. The moduli of vector bundles gives connections among different types of varieties, and sometimes yields new varieties that are difficult to describe by other means. The other is the linear system. In the same way as the classical construction of a map to a projective space, a vector bundle gives rise to a rational map to a Grassmannian if it is generically generated by its global sections. In this article, we shall describe some results for which vector bundles play such roles. They are obtained from an attempt to generalize Brill–Noether theory of special divisors, reviewed in Section 2, to vector bundles. Our main subject is rank-two vector bundles with canonical determinant on a curve C with as many global sections as possible: especially their moduli and the Grassmannian embeddings of C by them (Section 4). 1. Line bundles Let X be a smooth algebraic variety over the complex number field C. We consider the set of isomorphism classes of line bundles, or invertible sheaves, on X . This set enjoys two good properties, neither of which holds anymore Based on the author’s three talks given at JAMI in 1991, UCLA in 1992 and Durham University in 1993. Supported in part by a Grant under The Monbusho International Science Research Program: 04044081.