On the super Hilbert scheme of constant Hilbert polynomials

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 $k \neq 0$