Negative curves on algebraic surfaces

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of surfaces for which C^2 is not bounded below are in positive characteristic, and the general expectation is that no examples can arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field. The previous version of this paper claimed to give a counterexample to the Bounded Negativity Conjecture. The idea of the counterexample was to use Hecke translates of a smooth Shimura curve in order to create an infinite sequence of curves violating the Bounded Negativity Conjecture. To this end we applied Hirzebruch Proportionality to all Hecke translates, simultaneously desingularized by a version of Jaffee's Lemma which exists in the literature but which turns out to be false. Indeed, in the new version of the paper, we show that only finitely many Hecke translates of a special subvariety of a Hilbert modular surface remain smooth. This new result is based on work done jointly with Xavier Roulleau, who has been added as an author. The other results in the original posting of this paper remain unchanged.Comment: 14 pages, X. Roulleau added as author, counterexample to Bounded Negativity Conjecture withdrawn and replaced by a proof that there are only finitely many smooth Shimura curves on a compact Hilbert modular surface; the other results in the original posting of this paper remain unchange

Finite generation and geography of models

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.Engineering and Physical Sciences Research Council [grant number EP/H028811/1]; DFG-Forschergruppe 790 \Classi cation of Algebraic Surfaces and Compact Complex Manifolds", and by the OTKA Grants 77476 and 81203 of the Hungarian Academy of Sciences

Rationality of Seshadri constants and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture

International audienceIn this paper we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane

Computing Levi decompositions in Lie algebras

We consider the algorithmic problem of computing Levi decompositions in Lie algebras and Wedderburn–Malcev decompositions in associative algebras over the field of rational numbers. We propose deterministic polynomial time algorithms for both problems. The methods are based on the corresponding classical existence theorems. Computational experiences are discussed at the end of the paper

Newton–Okounkov Bodies of Exceptional Curve Plane Valuations Non-positive at Infinity

Producción CientíficaIn this note we announce a result determining the Newton–Okounkov bodies of the line bundle OP2 (1) with respect to exceptional curve plane valuations non-positive at infinityMinisterio de Economía, Industria y Competitividad (MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)Universitat Jaume I (grant P1-1B2015-02