Connectivity and the Nef Cone of the Hilbert Scheme of Hypersurfaces in the Grassmannian

In this thesis, we show that when $d \geq 3$, the Hilbert scheme $Hilb_{dT+1-\binom{d-1}{2}}(G(k,n))$ has 2 connected components, even though elements in both components have the same cohomology class. We used the tangent space of the Hilbert scheme and the Hartshorne's connectivity theorem and proved the correspondence between moduli space of plane curves and the Hilbert scheme. Applying Schubert calculus on this correspondence, we derived the number of components on the Hilbert scheme of the Grassmannian. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial $\binom{T+m}{m}-\binom{T+m-d}{m}$ in Grassmannian has at most 2 connected components, by generalizing plane curves as hypersurfaces. At last, we show the generators of the Nef cone of these Hilbert schemes, by observing curves in the Hilbert scheme which are dual to generators of N\'{e}ron-Severi group of the Hilbert scheme.