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The arithmetic and geometry of a class of algebraic surfaces of general type and geometric genus one

Citation

Lyons, Christopher Michael (2010) The arithmetic and geometry of a class of algebraic surfaces of general type and geometric genus one. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05272010-144845323

Abstract

We study of a class of algebraic surfaces of general type and geometric genus one, with a view toward arithmetic results. These surfaces, called CC surfaces here, have been classified over the complex numbers by Catanese and Ciliberto. At the heart of our work is a large monodromy result for a family containing all members of a large subclass of CC surfaces, called the admissible CC surfaces. This result is obtained by an analysis of degenerations of admissible CC surfaces. We apply this monodromy theorem to prove the Tate and semisimplicity conjectures for all admissible CC surfaces over finitely-generated fields of characteristic zero, which are statements about the Galois representations on their cohomology. We also apply the theorem to produce an example of an algebraic cycle on a Shimura variety of orthogonal type that is not contained in any proper special subvariety; this we do by using the period map of the aforementioned family. Finally, we deduce the existence of complex CC surfaces with the minimum possible Picard number.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:algebraic surfaces, surfaces of general type, Tate Conjecture, monodromy
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2010
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Graber, Thomas B.
  • Mantovan, Elena
  • Gholampour, Amin
Defense Date:25 May 2010
Record Number:CaltechTHESIS:05272010-144845323
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05272010-144845323
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5861
Collection:CaltechTHESIS
Deposited By: Christopher Lyons
Deposited On:03 Jun 2010 18:47
Last Modified:26 Dec 2012 03:26

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