Hilbert scheme and punctual Hilbert scheme

Abstract

In [16], Barbara Fantechi studied the restriction of the first higher direct image of the tangent sheaf of the Hilbert scheme of points on a smooth surface X along the Hilbert-Chow morphism on the subset of X(n) where exactly two points collide. She proved that for X being a surface of general type or a regular surface of Kodaira dimension 1, every deformation of X(n) and X[n] is induced naturally by deformation of X. In [20], Nigel Hitchin developed Barbara Fantechi's results. He showed that for a smooth rational projective surface X, there is a subspace of H1(X[n]; T) with the property that classes in this subspace are the contraction of Poisson structures on X[n] induced from Poisson structures on X with the class of the exceptional divisor of the Hilbert-Chow morphism. Therefore every class in this subspace is integrable by his unobstructed theorem on deformation. Especially, his results imply the unobstructedness of deformation of X[n] for X being a rigid smooth rational projective surface. Motivated by their results, we are going to study the higher direct image of the tangent sheaf of X[n] along the Hilbert-Chow morphism. By studying the higher direct image, we will establish some new results on unobstructedness of deformation of Hilbert scheme of points on smooth rational projective surfaces which can cover what Hitchin's results can imply.We will prove that the first higher direct image is scheme-theoretically supported on the singular locus of X(n) and we will give a nice construction of it which contains all the information of the first higher direct image. When n = 3, we will prove that the second higher direct image vanishes. These two results will lead us to the main theorem of this thesis: we will totally determine those smooth rational projective surfaces which satisfy the property that their Hilbert scheme of points has vanishing obstruction group. For the purpose of studying unobstructedness of the deformation of X[n], our results are strongerthan N. Hitchin's results in the sense that we can prove unobstructedness for more smooth rational projective surfaces.