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研究生： 陳家湘
Chen, Chia-Hsiang
論文名稱： 三次曲面上的直線及Del Pezzo曲面的分類
Lines on cubic surfaces and classification of Del Pezzo surfaces
指導教授： 卓士堯
Jow, Shin-Yao
口試委員: 陳俊成
Chen, Jiun-Cheng
陳正傑
Chen, Jheng-Jie
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2025
畢業學年度： 113
語文別： 英文
論文頁數： 37
中文關鍵詞： 三次曲面 、Del Pezzo曲面
外文關鍵詞： Cubic surface, Del Pezzo surface
相關次數： 點閱：304  下載：10 
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    • 三次曲面裡的一個經典問題是-光滑射影三次曲面定義在代數封閉體剛好包含二十七條$\mathbb{P}^3$裡的直線。在這篇文章裡我們使用現代手法證明這個結果並在一些條件下分類三次曲面。接著，我們介紹Del Pezzo曲面並在光滑射影且定義在代數封閉體時分類他們。最後，作為在三次曲面的二十七條線的推廣問題，我們研究Del Pezzo曲面上的$(-1)$曲線。

    A classical problem in the theory of cubic surfaces is that a smooth projective cubic surface over an algebraically closed field contains exactly twenty-seven lines in $\mathbb{P}^3$. In this paper, we provide a modern proof of this result and classify cubic surfaces under certain conditions. Next, we introduce Del Pezzo surfaces and classify them in the case of smooth, projective surfaces over an algebraically closed field. Finally, we generalize the problem of counting the twenty-seven lines on cubic surfaces by studying the (−1)-curves on Del Pezzo surfaces.

    Introduction 1 1 Background 2 1.1 Vector bundle and locally free sheaf . . . . 2 1.2 Divisors and sheaf associated to divisors . . . . 3 1.3 Morphism to Pn. .7 1.4 Complete linear systems and linear systems . . . . . . . . 7 1.5 Geometry on surface . . . . . 9 2 Cubic Surface in P3 13 2.1 Linear system with assigned base points . . .. . 13 2.2 Linear system of conics and cubics in P2 . . .. . . 15 2.3 A construction of smooth cubic surface in P3 and the 27 lines on them . . . . . . . . 19 2.4 Classification of smooth cubic surface in P3 . . . . . . 22 3 Del Pezzo surfaces 30 3.1 Definition of Del Pezzo surfaces and properties of them . . . . 30 3.2 Classification of Del Pezzo surface . . . . . 32 3.3 Numbers of (-1)-curves on Del Pezzo surfaces . . . . . . . 34 References 37

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