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Combinatorics and Geometry of Hilbert Schemes of Points on Surfaces

Cavey, Ian
http://rave.ohiolink.edu/etdc/view?acc_num=osu1681821198350912

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we study sections of line bundles on Hilbert schemes of points on surfaces. The Hilbert scheme of points on C2 has an algebraic description, due to Haiman, which allows for a concise description of its total coordinate ring. We use this to give an algebraic description of the total coordinate ring of the Hilbert scheme of points on any smooth projective toric surface. Motivated by the theory of Newton-Okounkov bodies, we then study degenerations of these rings to semigroup algebras, which are more suitable for combinatorial study. We give a combinatorial description of the resulting semigroup for the Hilbert scheme of points on C2, and a conjectural description for several other toric surfaces of interest. Our results imply upper bounds for the effective cones of these Hilbert schemes, which we conjecture to be sharp in several cases. Finally, we study the punctual Hilbert scheme parametrizing subschemes of C2 supported at a single point. Characters of these line bundles are known to be enumerated by higher Dyck paths via area and bounce. We give a combinatorial interpretation of the limit of these characters, the Duistermaat-Heckman measure of the punctual Hilbert scheme, in terms of similar statistics on continuous Dyck paths.
David Anderson (Advisor)
Eric Katz (Committee Member)
Maria Cueto (Committee Member)
88 p.
Mathematics
Hilbert schemes; toric varieties; Newton-Okounkov bodies; q,t-Catalan numbers

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Citations

  • Cavey, I. (2023). Combinatorics and Geometry of Hilbert Schemes of Points on Surfaces [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681821198350912

    APA Style (7th edition)

  • Cavey, Ian. Combinatorics and Geometry of Hilbert Schemes of Points on Surfaces. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1681821198350912.

    MLA Style (8th edition)

  • Cavey, Ian. "Combinatorics and Geometry of Hilbert Schemes of Points on Surfaces." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681821198350912

    Chicago Manual of Style (17th edition)

osu1681821198350912
213
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This open access ETD is published by The Ohio State University and OhioLINK.