Divisors on the blow-up of the projective plane
Let \(\mathbb{P}^2\) be the complex projective plane over the field of complex numbers, and \(\pi : S \to \mathbb{P}^2\) be the blow-up of \(\mathbb{P}^2\) at \(k\) generic points \(P_1, P_2, \dots, P_k\) with exceptional divisors \(E_1, E_2, \dots, E_k\). Then the Picard group of \(S\) is generated by \(H = \pi^* {\mathcal O} (1)\), \(E_1, \dots, E_k\). The author shows that if \(k\) and \(p > 2\) are positive integers, then the divisor \(pH - \sum^k_{i = 1} E_i\) is ample on \(S\) if and only if \(p^2 > k\). This gives an affirmative answer to a question of \textit{T. Fujita}.
scientific article; zbMATH DE number 733201