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研究生： 黎柏榮
Lai, Pak-Wing
論文名稱： 凱萊-巴拉赫定理
Cayley-Bacharach Theorems
指導教授： 卓士堯
Jow, Shin-Yao
口試委員: 蔡孟傑
Chuan, Meng Kiat
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： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 37
中文關鍵詞： 代數幾何 、代數曲線 、凱萊-巴拉赫定理
外文關鍵詞： Algebraic Geometry, Algebraic Curve, Cayley-Bacharach Theorem
相關次數： 點閱：50  下載：0 
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    • 本論文主要是研究凱萊-巴拉赫定理的不同證明及其推廣，從而了解經典代數幾何的一些主題，如獨立條件、馬克斯·諾特基本定理和除子。我們還介紹了一些有趣又相關的題材。

    The main propose is studying different proofs of Cayley-Bacharach theorem and its extension, thereby understanding some topics of classical algebraic geometry, such as independent conditions, Max Noether's fundamental theorem, and divisors. We also present some interesting related topics.

    1. Introduction---------------------------------------------5 2. Background-----------------------------------------------6 2.1 Algebraic Curves----------------------------------------6 2.2 Mapping between Affine and Projective planes------------8 2.3 Intersection of Curves----------------------------------9 2.4 Pappus’s Theorem---------------------------------------10 3. Algebraic structure of Curves---------------------------12 3.1 Linear System of Homogeneous Polynomials---------------12 3.2 Conditions Determined by Points------------------------14 3.3 Cayley–Bacharach Theorem-------------------------------18 4. Fundamental Theorem of Algebraic Functions--------------21 4.1 Conditions Determined by Intersection Points-----------21 4.2 Application of AF+BG Theorem---------------------------23 4.3 Related Theorems---------------------------------------25 5. Algebraic structure of Points---------------------------27 5.1 Divisors-----------------------------------------------27 5.2 Vector Space of Rational Functions---------------------30 5.3 CB4 - CB5----------------------------------------------32 6. Further Study-------------------------------------------36

    [1] Mark Green, David Eisenbud, and Joe Harris.Cayley-Bacharach Theoremsand Conjectures. Bulletin American Math. Society, 33 (1996), no.3, 295-324
    [2] W. Fulton,Algebraic Curves: An Introduction to Algebraic Geometryhttp://www.math.lsa.umich.edu/ wfulton/CurveBook.pdf
    [3] Casas Alvero, Eduardo.Algebraic Curves, the Brill and Noether way. 2019,Springer, ISBN: 978-3-030-29016-0
    [4] Hartshorne.Algebraic Geometry. 1977, Springer, ISBN: 978-1-4757-3849-0
    [5] Qingchun Ren, Jurgen Richter-Gebert and Bernd Sturmfels.Cayley-BacharachFormulas. The American Mathematical Monthly , Nov., 2015, Vol. 122, No. 9,pp. 845-854
    [6] Elena Anne Marchisotto.The Theorem of Pappus: A Bridge between Algebraand Geometry. The American Mathematical Monthly , Jun.-Jul., 2002, Vol.109, No. 6, pp. 497-516
    [7] A. Seidenberg.Pappus Implies Desargues. The American MathematicalMonthly , Mar., 1976, Vol. 83, No. 3, pp. 190-192

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