del Pezzo 曲面之幾何

Abstract

本文介紹del Pezzo曲面之研究。早期的研究主要以光滑曲面為對向，但近年則多考慮帶有奇點的曲面。因此第二章即討論各種奇點，始自第三章起正式定義del Pezzo 曲面，介紹光滑曲面的分類。第四章介紹Shokurov發展的complement 理論，並在第五章的weighted complete intersection 中給出例子。第六章介紹凱勒─愛因斯坦距離和del Pezzo曲面的關係。第七章與第八章是作者的研究結果利用黎曼─羅赫定理計算尤拉示性數並得到一種特別的不消沒定理。

The thesis in on the geometry of del Pezzo surfaces. Early researches focused on smooth surfaces, while recently surfaces with singularities have been mostly considered. Consequently, in Chapter 2, different types of singularities are first discussed, and then del Pezzo surfaces can be defined formally in Chapter 3. Research on smooth surfaces are also given there. In Chapter 4, we introduce the complement theory developed by Shokurov, and we give some examples of weighted complete intersection in Chapter 5. Chapter 6 is about the relation between Kahler-Einstein metrics and del Pezzo surfaces. In Chapter 7 and Chapter 8, we introduce our research result. We use Riemann-Roch theorem to calculated Euler characteristics, and then give a special type of nonvanishing theorem.