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.