On the influence of the Segre conjecture on the Mori cone of blown-up surfaces (Q2255097)

Given a smooth complex projective variety \(X\), the Mori Cone \(\overline{\mathrm{NE}}(X)\) of \(X\) is defined as the closure of the convex cone in \(N_1(X)\) spanned by the numerical classes of effective curves. Thanks to Mori's Cone theorem we know that the \(K_X\)-negative part of the cone is locally rationally polyhedral and it is spanned by extremal rays \(R_i\) [\textit{J. Kollár} and \textit{S. Mori}, Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics 134. Cambridge: Cambridge University Press (1998; Zbl 0926.14003)], \[ \overline{\mathrm{NE}}(X) = \overline{\mathrm{NE}}(X)|_{K_X \geq 0} + \sum R_i \] while very little is known about \(\overline{\mathrm{NE}}(X)|_{K_X \geq 0} := \{\alpha \in \overline{\mathrm{NE}}(X) | \alpha \cdot K_X \geq 0 \}.\) Some important conjectures on plane curves, such as the Nagata conjecture on linear systems on \(\mathbb{P}^2\), are strictly related to the shape of the Mori Cone of the blow-up of the plane at \(r\) general points. In this context, the equivalent conjectures of Segre, Harbourne, Gimigliano and Hirschowitz (SHGH) predict that if \(X = \mathrm{Bl}_r\mathbb{P}^2\) is the blow-up of the plane at \(r\) very general points, then a part of \(\overline{\mathrm{NE}}(X)|_{K_X \geq 0}\) is supported upon a spherical cone, i.e. is circular. A consequence on the geometry of \(X\) is the following decomposition of the Mori Cone, the so called (-1)-Curves Conjecture: if \(X\) is the blow up of \(\mathbb{P}^2\) at \(r\) very general points, then \[ \overline{\mathrm{NE}}(X) = \overline{\mathrm{Pos}}(X) + \sum R_i \] where the sum is taken over all the \(K_X\)-negative extremal rays \(R_i\) of \(\overline{\mathrm{NE}}(X)\) and the positive cone \(\overline{\mathrm{Pos}}(X)\) is defined in the following way: \[ \overline{\mathrm{Pos}}(X) = \{x \in N_1(X) | x^2 \geq 0, x \cdot h \geq 0\}. \] where \(h\) is an ample class. \textit{T. de Fernex} [in: Projective varieties with unexpected properties. A volume in memory of Giuseppe Veronese. Proceedings of the international conference ``Varieties with unexpected properties'', Siena, Italy, June 8--13, 2004. Berlin: Walter de Gruyter. 199--207 (2005; Zbl 1121.14006)], proved that the \((-1)\)-Curves Conjecture implies the Nagata Conjecture. In the spirit of the previous conjectures, one might hope that the all positive part of the Mori Cone of \(X\) is circular, i.e. the equality \[ \overline{\mathrm{NE}}(X)|_{K_X \geq 0} = \overline{\mathrm{Pos}}(X)|_{K_X \geq 0} \] holds. \textit{T. de Fernex} [``On the Mori cone of blow-ups of the plane'', Preprint, \url{arXiv:1001.5243v2}], proved that if \(X = \mathrm{Bl}_r\mathbb{P}^2\) is the blow-up of the plane at \(r\) very general points then the equality holds if \(r \leq 9\) and that if \(r \geq 10\) \[ \overline{\mathrm{NE}}(X)|_{(K_X - sL) \geq 0} = \overline{\mathrm{Pos}}(X)|_{(K_X - sL) \geq 0} \tag{1} \] where \(L\) is the preimage of the ample class on \(\mathbb{P}^2\) and the number \(s\) depends only on \(r\). In the paper under review the author proved a generalization of the latter result in the case of a blow-up of a smooth projective surface. He focuses on the study of curves with negative self intersection. He needs to formulate a generalization of the SHGH conjectures, in the formulation of Beniamino Segre, the Generalized Segre Conjecture, from which he derives a boundedness result, the List Conjecture, that plays the role of the \((-1)\)-Curves Conjectures in the case of \(\mathbb{P}^2\). If the List Conjecture is true then there exists an explicit \(s \in \mathbb{R}\) such that (1) holds.