Index intersection of ample divisors

Question

I'm trying to prove that the sum of two ample divisors on a projective complex algebraic surface S is it self an ample divisor.
To do this i need to verify that the index intersection between two ample divisors is positive.
Is it true that if A and B are two ample divisors on S the index intersection AB is a positive integer ?

Answer 1

I must say I've never heard the term "index intersection" before. Maybe this is an issue of language: the usual English phrase is "intersection number".

Anyway, yes, this is true. Here's the proof:

1. Intersection numbers are bilinear in both arguments, so we can assume $A$ and $B$ are very ample.
2. A very ample divisor is effective (by definition).
3. A very ample divisor has positive interesection number with any effective divisor (easy exercise).