Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory
 Title:
 Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory
 Author(s):
 Maddock, Zachary Alexander
 Thesis Advisor(s):
 de Jong, Aise Johan
 Date:
 2012
 Type:
 Theses
 Degree:
 Ph.D., Columbia University
 Department(s):
 Mathematics
 Persistent URL:
 https://doi.org/10.7916/D82B9568
 Abstract:
 This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anticanonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semistable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.
 Subject(s):
 Mathematics
 Item views
 816
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 Suggested Citation:
 Zachary Alexander Maddock, 2012, Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory, Columbia University Academic Commons, https://doi.org/10.7916/D82B9568.