希爾伯特概形之建構及其局部性質與應用

Abstract

本文分成兩個部分。第一個部分介紹希爾伯特概形與Hom概形的建構及建構所需的預備定理。第二部分探討一些Hom概形的局部結構：切空間的結構與維度估計，並運用這些結果來證明Mori的扳斷引理及一個有關Fano簇上的有理曲線之存在性定理。

There are two parts in this thesis. The first part consists of constructions of Hilbert schemes and Hom schemes with their preliminaries. The second part is some local properties of Hom schemes, structure of its tangent spaces and estimation of its dimension, and their applications: proving Mori's bend-and-break lemmas and a theorem about the existence of rational curves on Fano varieties.