Constructing new ample divisors out of old ones (Q1974922)

The author studies the ample cone of surfaces \(S\) obtained by blowing-up \(\mathbb{P}^2\) at \(k\) distinct points \(p_1,\dots,p_k\). A vector \((d;m_1, \dots, m_k)\in \mathbb{Z}_+ \times \mathbb{Z}^k_{\geq 0}\) is said to be ample (respectively nef) if there exists a surface \(S\) as above such that \(\pi^*{\mathcal O}_{\mathbb{P}^2} (d)-\sum^k_{j=1} m_jE_j\) is ample (respectively nef), where \(\pi\) is the blow-up and \(E_j=\pi^{-1}(p_j)\), \(j=1,\dots,k\). The following gluing theorem is proved. Let \((d;m_1, \dots, m_k,m)\) be an ample (respectively nef) vector and let \((m; \alpha_1, \dots,\alpha_n) \in\mathbb{Z}_+^{n+1}\) be a nef vector; then \((d;m_1, \dots, m_k,\alpha_1, \dots, \alpha_n)\) is ample (respectively nef). This theorem, combined with the action of the Cremona group on the ample cone gives rise to an algorithmic procedure for detecting new ample classes in the Picard group of rational surfaces. Moreover the author associates a Seshadri-like constant \({\mathcal R}_k({\mathcal L})\), called remainder, to any ample line bundle \({\mathcal L}\) and the integer \(k\). These constants, which are invariant under rescaling of \({\mathcal L}\), allow him to rephrase a classical conjecture of Nagata on the ampleness of \(\pi^* {\mathcal O}_{\mathbb{P}^2} (d)-m\sum^k_{j=1} E_j\) in the following very simple way: \({\mathcal R}_k ({\mathcal O}_{\mathbb{P}^2} (1))=0\) for every \(k\geq 9\). By using the algorithm above the author establishes several results towards Nagata's conjecture and proposes a weaker conjecture on the bound of \({\mathcal R}_k({\mathcal O}_{\mathbb{P}^2}(1))\) in terms of continued fractions approximations of \(\sqrt k\). He also generalizes related results by \textit{Geng Xu} [Manuscr. Math. 86, No. 2, 195-197 (1995; Zbl 0836.14004)] and \textit{O. Küchle} [Math. Ann. 304, 151-155 (1996; Zbl 0834.14024)]. In a final section the meaning of the remainders \({\mathcal R}_k({\mathcal L})\) is explained in terms of volume of a symplectic manifold and the part of it which can be filled by a packing of \(k\) balls.