Titlebook: Birational Geometry, Rational Curves, and Arithmetic; Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel Book 2013 Springer Science+Business M

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Robert Ebermann,Ibrahim ElmadfaThis survey is an invitation to recent developments in higher dimensional birational geometry.

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Curves of Low Degrees on Fano Varieties,We survey the period maps of some Fano varieties and the geometry of their spaces of curves of low genera and degrees.

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Uniruledness Criteria and Applications,We discuss uniruledness criteria on higher-dimensional varieties and their applications.

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The Cone of Curves of K3 Surfaces Revisited,The following theorem was proved in [4] over the complex numbers. It turns out that the proof given there works with very small adjustments in arbitrary characteristic.

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On the Ubiquity of Twisted Sheaves,We describe some recent work on the uses of twisted sheaves in algebra, arithmetic, and geometry. In particular, we touch on the role of twisted sheaves in:

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Birational Geometry, Rational Curves, and Arithmetic978-1-4614-6482-2Series ISSN 2365-9564 Series E-ISSN 2365-9572

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https://doi.org/10.1007/978-3-211-49348-9miliar with characteristic-.-geometry but who would like to see similarities, as well as differences, to complex geometry. More precisely, these notes are on algebraic surfaces in positive characteristic and assume familiarity with the complex side of this theory, say, on the level of Beauville’s book [9].

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