# Index intersection of ample divisors

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 ?

1. Intersection numbers are bilinear in both arguments, so we can assume $A$ and $B$ are very ample.
• @dario: an equivalent definition of very ample is this: $A$ is very ample if there is an emedding $X \hookrightarrow \mathbf P^n$ such that $A= H_{|X}$, where $H$ is the hyperplane class on $\mathbf P^n$. If you look into how the "associated morphism" is defined, you'll see that's the same thing. – user64687 Sep 19 '14 at 14:25