Effective Divisor Classes on a Ruled Surface

The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron-Severi group. Despite the finite generation of the whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, and the methods used to explicitly calculate this monoid seem to vary widely as one proceeds from one type of surface to another in the standard classification scheme (see Rosoff, 1980, 1981). In this paper we shall use concrete vector bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled surface over a given base curve. We will find that, over a base curve of genus 0, the monoid of effective divisor classes is very simple, having two generators (which is perhaps to be expected), while for a ruled surface over a curve of genus 1, the monoid is more complicated, having either two or three generators. Over a base curve of genus 2 or greater, we will give necessary and sufficient conditions for a ruled surface to have its monoid of effective divisor classes finitely generated; these conditions point to the existence of many ruled surfaces over curves of higher genus for which finite generation fails.