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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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WRAP_THESIS_Brown_1995.pdf - Submitted Version Download (35Mb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b1399911~S1
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 or Dissertation (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Mappings (Mathematics), Torus (Geometry) | ||||
| Official Date: | June 1995 | ||||
| Dates: |
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| 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| Publication Status: | Unpublished | ||||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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