Birational maps of surfaces with invariant curves.

Doctoral Dissertation


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.


Attribute NameValues
  • etd-07222005-101317

Author Daniel Robert Jackson
Advisor Jeff Diller
Contributor Andrew Sommese, Committee Member
Contributor Liviu Nicolaescu, Committee Member
Contributor Karen Chandler, Committee Member
Contributor Jeff Diller, Committee Chair
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Defense Date
  • 2005-07-01

Submission Date 2005-07-22
  • United States of America

  • birational maps

  • algebraic geometry

  • dynamical systems

  • invariant curves

  • University of Notre Dame

  • English

Record Visibility Public
Content License
  • All rights reserved

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