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 Hilb1∣1(ΠOP1(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=0 and vanishing for k=0. Moreover, since the odd dimension of this Hilbert scheme is 2, we can see that Hilb2∣1(ΠV) is projected for k=0 and not projected for all k=0