Vector Bundles over Ruled Surfaces
In this chapter and in Chapter 8 we will discuss the structure of stable bundles on special classes of surfaces, ruled and elliptic surfaces. In general, it seems a little difficult to obtain detailed information about moduli spaces when stability is defined with respect to an arbitrary ample divisor. We will need instead to consider ample divisors adapted to the geometry of the surfaces at hand. The examples we have in mind are all given as fibrations over a base curve, and the fibration itself will be the most interesting geometric feature of the surface. Thus, we will try to consider ample divisors which reflect the fibration. The class of a fiber is not ample; it lives at the boundary of the ample cone. Instead we shall consider ample divisors which are sufficiently close to the class of a fiber, where how close will depend on the particular choice of Chern classes of the problem we want to study. We will further study bundles on the surface X by studying their restrictions to the fibers of the fibration. This method of studying bundles on a surface by looking at their restrictions to curves on the surface is one which has been successfully applied in a wide variety of contexts.
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