The ABC conjecture is a number-theoretic analogue of the Mason-Stothers theorem from polynomial algebra. Specifically, let S(n) denote the �conductor� of the integer n, i.e. the product of the distinct prime factors of the integer n. The ABC conjecture claims that, given any power r > 1, there is a corresponding constant K, depending only on r, such that for any relatively prime integers a, b and c with a + b = c
Max {|a|, |b|, |c|} < K [S(abc)] r
It has a number of amazing consequences, such as finiteness of the number of Mersene primes, and an �asymptotic� Fermat�s Last Theorem.
The Neron-Severi group of an algebraic surface is often not difficult to calculate, and in particular its rank has classically been studied as a key invariant of such surfaces. A more difficult problem is to determine those algebraic equivalence classes in the group that can be represented by effective divisors; these classes form a monoid in the Neron-Severi group, which naturally generates the whole group. Despite the (classically known) finite generation of the Neron-Severi group, the monoid of effective divisor classes need not be finitely generated. The calculation of this monoid is often difficult, and the techniques for its calculation seem to vary widely as one moves between the standard surface types in the Kodaira classification scheme of algebraic surfaces.
Here are just a few examples (in what follows we will let X denote the algebraic surface under consideration, and M(X) its monoid of effective divisor classes in the Neron-Severi group of X):
X: Projective plane blown up at eight or fewer points: M(X) finitely generated. For nine points or more, finite generation fails.
X: Abelian variety (or higher-dimensional tori): M(X) is �usually� finitely generated, but finite genration fails in even some simple examples, such as when X is the product of an elliptic curve with itself.
X: A ruled surface over a base curve of genus g. If g = 0 or 1, M(X) is generated by either 2 or 3 generators. For ruled surfaces over curves of higher genus, finite generation may or may not occur, depending on the 2-bundle of which X is the projectivization.