On the super Hilbert scheme of constant Hilbert polynomials

Algebraic geometry;Supergeometry

Jang, Mi Young

关键词: Algebraic geometry;    Supergeometry;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/98128/JANG-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 $k \neq 0$.