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Even sets of ($-4$)-curves on rational surface

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Martí Sánchez, M. Osaka J. Math. 48 (2011), 675–690 EVEN SETS OF (�4)-CURVES ON RATIONAL SURFACE MARÍA MARTÍ SÁNCHEZ (Received August 10, 2009, revised February 26, 2010) Abstract We study rational surfaces having an even set of disjoint (�4)-curves. The prop- erties of the surface S obtained by considering the double cover branched on the even set are studied. It is shown, that contrarily to what happens for even sets of (�2)-curves, the number of curves in an even set of (�4)-curves is bounded (less or equal to 12). The surface S has always Kodaira dimension bigger or equal to zero and the cases of Kodaira dimension zero and one are completely characterized. Several examples of this situation are given. 1. Introduction Let X be a smooth surface. A set of � disjoint smooth rational curves N1, : : : , N� is called an even set if there exists L 2 Pic(X ) such that 2L � N1 C � � � C N� . In this note we study even sets of curves N1, : : : , N� where each Ni is a (�4)-curve (i.e. a smooth rational curve with self-intersection �4) on rational surfaces. We prove that, contrarily to what happens for even sets of (�2)-curves (cf. [6]), the number of curves in an even set of (�4)-curves is bounded. More precisely we show that the maximal number of curves in such a set is 12. Given an even set of smooth rational curves one can consider the double cover branched on these curves. For even sets of (�2)-curves on rational surfaces, such a double cover is again a rational surface (see [6]). In contrast again the double cover of a rational surface branched on an even set of (�4)-curves has always Kodaira dimension � 0. In this paper we characterize completely the even sets of (�4)-curves on rational surfaces, such that the corresponding double cover has Kodaira dimension 0 or 1. More precisely we show that any even set of (�4)-curves on a rational surface, whose corresponding double cover has Kodaira dimension 0 or 1, are components of fibres of a not relatively minimal elliptic fibration. We gi

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