Log geometry and extremal contractions
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2015
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| Online Access: | http://hdl.handle.net/1721.1/99064 |
| _version_ | 1826211571789987840 |
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| author | Svaldi, Roberto |
| author2 | James McKernan. |
| author_facet | James McKernan. Svaldi, Roberto |
| author_sort | Svaldi, Roberto |
| collection | MIT |
| description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. |
| first_indexed | 2024-09-23T15:08:06Z |
| format | Thesis |
| id | mit-1721.1/99064 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T15:08:06Z |
| publishDate | 2015 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/990642019-04-10T08:47:12Z Log geometry and extremal contractions Svaldi, Roberto James McKernan. Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics. Mathematics. Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. Cataloged from PDF version of thesis. Includes bibliographical references (pages 77-82). The Minimal Model Program (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic variety X, [Delta] it is allowed to change the variety under scrutiny as long as its field of rational functions remains the same. In this thesis we study two problems that are inspired by the techniques developed in the last 30 years by various mathematicians in an attempt to realize the Minimal Model Program for varieties of any dimension. In the first part of the thesis, we prove a result about the existence and distribution of rational curves in projective algebraic varieties. We consider projective log canonical pairs (X,[Delta] A) where the locus Nklt(X,[Delta] A) of maximal singularities of the pair (X,[Delta] A) is nonempty. We show that if Kx[Delta]+ A is not nef then there exists an algebraic curves C, whose normalization is isomorphic to A1, contained either in X \ Nklt(X,[Delta] A) or in certain locally closed varieties that stratify Nklt(X,[Delta] A). This result implies a strengthening of the Cone Theorem for log canonical pairs. In the second part, we study certain varieties that naturally arise as possible outcomes of the classification algorithm proposed by the MMP. These are called Mori fibre spaces. A Mori fibre space is a variety X with log canonical singularities together with a morphism f : X --> Y such that the general fiber of f is a positive dimensional Fano variety and the monodromy of f is as large as possible. We show that being the general fiber of a Mori fiber space is a very restrictive condition for Fano varieties with terminal Q-factorial singularities. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We apply our criteria to figure out what Fano varieties satisfy this property up to dimension three and to study the case of certain homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated. by Roberto Svaldi. Ph. D. 2015-09-29T19:01:09Z 2015-09-29T19:01:09Z 2015 2015 Thesis http://hdl.handle.net/1721.1/99064 921852296 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 82 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Svaldi, Roberto Log geometry and extremal contractions |
| title | Log geometry and extremal contractions |
| title_full | Log geometry and extremal contractions |
| title_fullStr | Log geometry and extremal contractions |
| title_full_unstemmed | Log geometry and extremal contractions |
| title_short | Log geometry and extremal contractions |
| title_sort | log geometry and extremal contractions |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/99064 |
| work_keys_str_mv | AT svaldiroberto loggeometryandextremalcontractions |