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Flips in low codimension : classification and quantitative theory

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Brown, Gavin Dennis (1995) Flips in low codimension : classification and quantitative theory. PhD thesis, University of Warwick.

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Abstract

A flip is a birational map of 3-folds X- ---> X+ which is an isomorphism away from curves C- c X- and C+ c X+ and does not extend across these curves. Flips are the primary object of study of this thesis. I discuss their formal definition and history in Chapter 1.

Flips are well known in toric geometry. In Chapter 2, I calculate how the numbers K3 and χ(nK) differ between X- and X+ for toric flips. These numbers are also related in a primary way by Riemann-Roch theorems but I keep that quiet until Chapter 5.

In Chapter 3, I describe a technique, which I learned from Miles Reid, for constructing a flip as C* quotients of a local variety 0 E A, taken in different ways. The codimension of my title refers to the minimal embedding dimension of 0 E A. The case of codimension 0 turns out to be exactly the case of toric geometry as studied in Chapter 2. The main result of Chapter 3 classifies the cases when A c C5 is a singular hypersurface, that is, when A defines a flip in codimension 1.

Chapters 4 and 5 concern themselves with computing new examples of flips in higher codimension and studying changes in general flips. I indicate one benefit of knowing how these changes work.

The main results of Chapters 2 and 3 have been circulated informally as [2] and [3] respectively.

Item Type: Thesis (PhD)
Subjects: Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH): Mappings (Mathematics), Torus (Geometry)
Official Date: June 1995
Dates:
DateEvent
June 1995Submitted
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Supervisor(s)/Advisor: Reid, Miles (Miles A.)
Sponsors: Engineering and Physical Sciences Research Council (EPSRC)
Extent: v, 102 p.
Language: eng

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