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Birational maps of surfaces with invariant curves.

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posted on 2005-07-22, 00:00 authored by Daniel Robert Jackson
We study curves that are invariant under a birational map f:X->X of a complex projective surface X. We show that if X is a minimal rational surface and f is an algebraically stable (AS) map with first dynamical degree larger than one, then any invariant curve for f has arithmetic genus at most 1. In particular, invariant curves for AS birational maps of the projective plane must have degree 3 or less. Next we find formulas for all of the AS quadratic birational maps of the projective plan whose indeterminacy is constrained to lie on an invariant curve Q; however, we exclude the cases when Q is an irreducible curve of genus 1. Finally we study the dynamics of some of these quadratic maps. By studying the induced real maps of the real projective plane we find a class of maps exhibiting maximal entropy in its real dynamics. Also we present an example in which our strategy fails to find such a map.

History

Date Modified

2017-06-02

Defense Date

2005-07-01

Research Director(s)

Jeff Diller

Committee Members

Andrew Sommese Liviu Nicolaescu Karen Chandler

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

etd-07222005-101317

Publisher

University of Notre Dame

Program Name

  • Mathematics

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