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Type of Document Dissertation Author Jackson, Daniel Robert Author's Email Address DoctorDanielJackson@excite.com URN etd-07222005-101317 Title Birational maps of surfaces with invariant curves. Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Jeff Diller Committee Chair Andrew Sommese Committee Member Karen Chandler Committee Member Liviu Nicolaescu Committee Member Keywords
- algebraic geometry
- dynamical systems
- invariant curves
- birational maps
Date of Defense 2005-07-01 Availability unrestricted Abstract 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.
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