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 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.
 目錄Contents口試委員審定書i誌謝ii摘要iiiAbstract iv1 Introduction 11.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . 22 Singularities 22.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Log singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Toric varieties and singularities . . . . . . . . . . . . . . . . . . . . . 132.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Del Pezzo Surfaces 194 Complements on Log Surfaces 214.1 n-complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Weighted Complete Intersection 285.1 Weighted projective space . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Weighted complete intersection . . . . . . . . . . . . . . . . . . . . . 306 Kahler-Einstein Metric 357 Euler Characteristics 377.1 Singular Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . 377.2 Euler characteristics under L-blowups . . . . . . . . . . . . . . . . . . 408 Nonvanishing 43Reference 46
 [1] G. Belousov “The maximal number of singular points on log del Pezzo surfaces”,J. Math. Sci. Univ. Tokyo.16, 231-238 (2009)[2] I. Cheltsov, “Log canonical thresholds on hypersurfaces”, Sbornik: Mathematics192(2001),1241-1257.[3] I. Cheltsov, D. Kosta, “ Computing αinvariants of Singular Del Pezzo Surfaces”,arXiv:1010.0043.[4] Ivan Cheltsov, Constantin Shramov, “Del Pezzo Zoo”, arXiv:0904.0114.[5] J. A. Chen, M. Chen, “An optimal boundedness on weak Q-Fano threefolds”,Adv. Math., 219, (2008), 2086-2104. arXiv 0712.4356.[6] D. F. Coray, M. A. Tsfasman, “Arithmetic on singular Del Pezzo surfaces”.Proc. London Math. Soc (3), 57(1) , 25–87 (1988).[7] I. Dolgachev, “Weighted projective spaces”.[8] A. R. Fletcher, “Working on Weighted Complete Intersections”, L.M.S. LectureNote Series 281 (2000), 101-173.[9] W. Fulton, “Introduction to Toric Varieties”, Princeton University Press, 1993.[10] Robin Hartshorne, “Algebraic Geometry”, Springer GTM 52.[11] Y. Kawamata, K. Matsuda, and K. Matsuki, “Introduction to the MinimalModel Problem”, Algebraic Geometry, Sendai 1985, Advanced Studies in PureMath. 10, (1987) Kinokuniya and North-Holland, 283-360.[12] Janos Kollar, “Flips and Abundance for Algebraic Threefolds” (Salt Lake City,UT, 1991), Asterisque 211, Soc. math. France. Montronge, 1992.[13] Janos Kollar, “Singularities of Pairs”, Proceedings of Symposia in Pure Mathematics62 (1997), 221-287.[14] J. Kollar, S. Mori , ‘Birational Geometry of Algebraic Varieties”, CambridgeUniversity Press, 1998.[15] J. Neukirch, “Algebraic Number Theory”, Springer Comprehensive studies 322.[16] Yu. G. Prokhorov.“ Lectures on complements on log surfaces”, arXiv:9912111.[17] Yu. G. Prokhorov and A.B. Verevkin. “The Riemann-Roch theorem on surfaceswith log-terminal singularities” J. Math. Sci., 140, No.2 (2007).[18] M. Reid, “Young person’s guide to canonical singularities” Proc. Symp. PureMath., 46, 343-416 (1987).[19] V. V. Shokurov, “Complements on surfaces” J. Math. Sci. , 102,No. 2 (2000).[20] G. Tian, “On Kahler-Einstein metrics on certain Kahler manifolds with c(M) >0 “. Invent, math. 89 (1987) 225-246.[21] V. Tosatti, “Kahler-Einstein metrics on Fano surfaces”,Expositiones Mathematicae30, 11-30 (2012).
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