# Ample invertible sheaves on projective surfaces

This question popped up as I was thinking about intersection theory on fibered surfaces, which is slightly different from the situation I'm about to describe, and as a result I may have gotten some facts wrong. Anyway, here goes:

Let $X$ be a regular projective surface. If $D$ be an ample effective divisor on $X$, corresponding to an ample invertible sheaf $\mathcal{O}_X(D)$ on $X$. Then, for every closed curve $C\subset X$, $\mathcal{O}_X(D)|_C$ is ample on $C$, and hence has positive degree as an invertible sheaf on $C$. In terms of intersection theory, this means that $D\cdot C > 0$ for every closed curve $C\subset X$.

But...isn't the intersection pairing negative semidefinite? Hence, $D\cdot D\le 0$. At first, this seems like it would break the ampleness of $D$, and hence regular projective surfaces $X$ can't have any ample divisors, but this is plainly false since you can just take $X = \mathbb{P}^2$, and $D$ to be any prime divisor.

Where is the problem(s) with my logic??

• The intersection pairing is certainly not negative semidefinite. Think of intersection theory on $\Bbb P^2$. Some curves may have negative self-intersection but a typical divisor won't. Commented Apr 23, 2017 at 5:35
• @LordSharktheUnknown Hmm, on $\mathbb{P}^2$, is it true that the intersection number of a curve of degree $d$ and a curve of degree $e$ is $d\cdot e$? So, the self intersection of a curve of degree $d$ is $d^2$? Interesting...what about fibered surfaces causes the intersection pairing to be negative semidefinite?
– user355183
Commented Apr 23, 2017 at 6:42
• Fibered or not, the intersection pairing is not negative semi-definite. Where did you read that? Commented Apr 23, 2017 at 13:12
• @Mohan Hmm, well on fibered surfaces usually the intersection pairing is only defined when one of the divisors is vertical, and so you can only talk about the self intersection of vertical divisors, and restricted to vertical divisors, the intersection pairing is negative semidefinite. (This is Qing Liu's book, chapter 9.1)
– user355183
Commented Apr 23, 2017 at 21:26
• I do not know what you mean. If $X$ is a regular projective surface, there is an intersection pairing on it, whether it is fibered or not. If fibered over a projective curve, it still has horizontal divisors and by definition of projective surface, has an ample divisor which intersects all curves positively. Commented Apr 23, 2017 at 23:14