# 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 ?

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).
• But on my book a divisor is very ample if the associated morphism from my surface to the n-dimensional complex projective space is an embedding.(i'm not assuming that a very ample divisor is effective) is it a consequence of the previous definition? Sep 19, 2014 at 14:20
• I've prooved the index intersection between an ample divisor and an effective divisor is positive. The proble is to change the effective divisor with another ample divisor. Sep 19, 2014 at 14:24
• @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, 2014 at 14:25
• ok now i understand- Thanks Sep 19, 2014 at 14:30