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Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications
Miller, Jason A

2014, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis draws a connection between two areas of algebraic geometry, spherical varieties and Okounkov bodies, in order to study the structure of Borel orbit closures in wonderful group compactifications. Spherical varieties are a natural generalization of many classes of varieties equipped with group actions such as flag varieties, symmetric varieties, and toric varieties. The theory of Okounkov bodies is a fascinating recent development generalizing the polytopes that appear in toric geometry to any projective algebraic variety.

Let X be a projective spherical G-variety equipped with a very ample G-line bundle L. Choosing a reduced decomposition of the longest element of the Weyl group determines a valuation vN on the ring of sections, R(X,L). One can then use Okounkov theory to encode information about the G-orbits of a spherical variety in terms of the associated Newton polytope. Each G-orbit closure of X determines a face of the Newton polytope. This correspondence allows one to use the combinatorial methods of convex geometry to answer questions about the G-orbit closures of the spherical variety X. However for nontoric spherical varieties, the G-orbit structure is too coarse-grained. A great deal of information about the spherical variety, such as the intersection theory, is determined by the structure of the Borel orbits.

In this thesis we consider wonderful group compactifications. We prove that one can extend the correspondence between G-orbits and faces to the Borel orbits for this class of varieties. Given any Borel orbit closure of a wonderful group compactification, we show that the Okounkov construction will yield a finite union of faces of the Newton polytope. This correspondence can be shown to enjoy many of the same nice properties as in the case of G-orbits: the dimension of the space of global sections of L is given by the number of lattice points in the union of faces, and the degree of any Borel orbit closure is the sum of the normalized volumes of the associated faces.
Gary Kennedy, Ph.D. (Advisor)
Roy Joshua, Ph.D. (Committee Member)
James Cogdell, Ph.D. (Committee Member)
111 p.

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Miller, J. (2014). Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Miller, Jason. "Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications." Electronic Thesis or Dissertation. Ohio State University, 2014. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Miller, Jason "Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications." Electronic Thesis or Dissertation. Ohio State University, 2014. https://etd.ohiolink.edu/

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