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 Title: On the super Hilbert scheme of constant Hilbert polynomials Author(s): Jang, Mi Young Director of Research: Katz, Sheldon Doctoral Committee Chair(s): Nevins, Thomas Doctoral Committee Member(s): Haboush, William; Schenck, Henry Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Algebraic Geometry, Supergeometry Abstract: In this thesis we mainly consider supermanifolds and super Hilbert schemes. In the first part of this dissertation, we construct the Hilbert scheme of $0$-dimensional subspaces on dimension $1 | 1$ supermanifolds. By using a flattening stratification, we compute the local defining equation for the super Hilbert scheme. From local defining equations, we conclude that the Hilbert scheme of constant Hilbert polynomials on dimension $1| 1$ supermanifolds is smooth. The second part of this thesis concerns the smoothness and the non smoothness of $0$-dimensional subspaces on some supermanifolds of higher dimensions, which is related with the future study chapter. The last part is devoted to the splitness of the Hilbert scheme. The non-splitness of supermanifolds can be deduced from the non vanishing of some cohomology class, called the obstruction class. We find examples of both split and non-split super Hilbert schemes. For the split case, we find a split model which is isomorphic to $Hilb^{1|1}(\Pi O_{\mathbb{P}^1}(k))$ for any $k$. For the non-split case, we compute the obstruction class of the super Hilbert scheme $Hilb^{2|1}( \Pi \oo_{\mathbb{P}^1}(k) )$ and show that this class is not vanishing for $k \neq 0$ and vanishing for $k=0$. Moreover, since the odd dimension of this Hilbert scheme is 2, we can see that $Hilb^{2|1}(\Pi V)$ is projected for $k=0$ and not projected for all $k \neq 0$. Issue Date: 2017-07-10 Type: Thesis URI: http://hdl.handle.net/2142/98128 Rights Information: Copyright 2017 Mi Young Jang Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
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