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.
