Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 243-248). / (cont.) yield a new proof of a result of Mochizuki yield a new proof of a result of Mochizuki Frobenius-unstable bundles for C general, and hence obtaining a self-contained proof of the resulting formula for the degree of V₂. / Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P¹ with ramification to order e[sub]i at general points P[sub]i the case that all e[sub]i are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in positive and mixed characteristics, yielding a result on lifting linear series from characteristic p to characteristic 0, and even showing promise for generalization to higher-dimensional varieties. Now, let C be a curve of genus 2 over a field k of positive characteristic, and V₂ the Verschiebung rational map induced by pullback under Frobenius on moduli spaces of semistable vector bundles of rank two and trivial determinant. We show that if the Frobenius-unstable vector bundles are deformation-free in a suitable sense, then they are precisely the undefined points of V₂, and may each be resolved by a single blow-up; in this setting, we are able to calculate the degree of V₂ in terms of the number of Frobenius-unstable bundles, and describe the image of the exceptional divisors. We finally examine the Frobenius-unstable bundles on C by studying connections with vanishing p-curvature on certain unstable bundles on C. Using explicit formulas for p-curvature, we completely describe the Frobenius-unstable bundles in characteristics 3, 5, 7. We classify logarithmic connections with vanishing p-curvature on vector bundles of rank 2 on P¹ in terms of self-maps of P¹ with prescribed ramification. Using our knowledge of such maps, we then glue the connections to a nodal curve and deform to a smooth curve to / by Brian Osserman. / Ph.D.
| Identifer | oai:union.ndltd.org:MIT/oai:dspace.mit.edu:1721.1/28310 |
| Date | January 2004 |
| Creators | Osserman, Brian, 1977- |
| Contributors | Johan de Jong., Massachusetts Institute of Technology. Dept. of Mathematics., Massachusetts Institute of Technology. Dept. of Mathematics. |
| Publisher | Massachusetts Institute of Technology |
| Source Sets | M.I.T. Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
| Format | 248 p., 16120226 bytes, 16152823 bytes, application/pdf, application/pdf, application/pdf |
| Rights | 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 |
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