Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic

The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and...

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Main Author: Fanelli, Andrea
Other Authors: Cascini, Paolo
Published: Imperial College London 2015
Subjects:
510
Online Access:http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.666503
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spelling ndltd-bl.uk-oai-ethos.bl.uk-6665032016-08-04T03:44:09ZTwo structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristicFanelli, AndreaCascini, Paolo2015The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and the minimal model program. We ask which Fano varieties can appear as a fibre of a Mori fibre space and introduce the notion of fibre-likeness to study this property. This turns out to be a rather restrictive condition: in order to detect this property, we obtain two criteria (one sufficient and one necessary), which turn into a characterisation in the rigid case. Many applications are discussed and the basis for the classification of fibre-like Fano varieties is presented. In the second part of the thesis, an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic is provided: this gives an effective bound on the multiple which makes an ample line bundle D very ample. A careful study of pathological surfaces is presented here in order to bypass the classical cohomological approach. As a consequence, we obtain a Kawamata-Viehweg-type vanishing theorem for arbitrary smooth algebraic surfaces in positive characteristic.510Imperial College Londonhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.666503http://hdl.handle.net/10044/1/26285Electronic Thesis or Dissertation
collection NDLTD
sources NDLTD
topic 510
spellingShingle 510
Fanelli, Andrea
Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
description The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and the minimal model program. We ask which Fano varieties can appear as a fibre of a Mori fibre space and introduce the notion of fibre-likeness to study this property. This turns out to be a rather restrictive condition: in order to detect this property, we obtain two criteria (one sufficient and one necessary), which turn into a characterisation in the rigid case. Many applications are discussed and the basis for the classification of fibre-like Fano varieties is presented. In the second part of the thesis, an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic is provided: this gives an effective bound on the multiple which makes an ample line bundle D very ample. A careful study of pathological surfaces is presented here in order to bypass the classical cohomological approach. As a consequence, we obtain a Kawamata-Viehweg-type vanishing theorem for arbitrary smooth algebraic surfaces in positive characteristic.
author2 Cascini, Paolo
author_facet Cascini, Paolo
Fanelli, Andrea
author Fanelli, Andrea
author_sort Fanelli, Andrea
title Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
title_short Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
title_full Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
title_fullStr Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
title_full_unstemmed Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic
title_sort two structural aspects in birational geometry : geography of mori fibre spaces and matsusaka's theorem for surfaces in positive characteristic
publisher Imperial College London
publishDate 2015
url http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.666503
work_keys_str_mv AT fanelliandrea twostructuralaspectsinbirationalgeometrygeographyofmorifibrespacesandmatsusakastheoremforsurfacesinpositivecharacteristic
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