A divisor \(D\) on a smooth projective surface \(X\) is ample if and only if \[ D^2 > 0 \quad\text{and}\quad D \cdot C > 0 \text{ for every irreducible curve } C \subseteq X . \]
This is the surface analogue of “ample means positive degree” on a curve, and the two conditions are exactly the two dimensions of subvariety: positivity against every curve, and positivity against the surface itself.
Both are needed. On the blowup of \(\PP^2\) at a point, \(D = \pi^*H\) has \(D^2 = 1 > 0\) but \(D \cdot E = 0\) on the exceptional curve, so it is not ample — it is only the pullback of an ample class, and it contracts \(E\). Conversely a class can meet every curve positively on a surface while failing \(D^2 > 0\), which is why the first condition is not implied by the second.
The criterion is purely numerical, so ampleness on a surface depends only on the class in \(\NS(X)\). That is what lets one draw the ample cone inside \(\NS(X) \otimes \RR\) and use the Hodge index theorem to describe it.
Dependencies
D-SRFINTThe intersection pairing on a surfaceuses