代數幾何 (書)

代數幾何（Algebraic Geometry）係一本好有影響力嘅代數幾何教科書，喺1977年由Robin Hartshorne寫，Springer-Verlag出版。

《Algebraic Geometry》

作者	Robin Hartshorne
語言	英文
題材	代數幾何
體裁	教科書
出版	Springer-Verlag
出版時間	1977
頁數	496
ISBN	0-387-90244-9

影響力

佢係第一本寫畀研究生睇，詳細介紹概形論嘅教科書。喺香港嚟講，科大李衛平教授教嘅代數幾何課程都係用呢本書。

內容

第一課標題係「簇」，主要講經典嘅代數封閉場上嘅代數簇理論，呢課用咗好多交換代數入面嘅定理，例如Hilbert's Nullstellensatz，主要嘅參考書係Atiyah-Macdonald、Matsumura同Zariski-Samuel。第二同第三課分別係「概形」同「上同調」，係呢本書嘅重點。最後兩課「曲線」同「曲面」就係用第二三課教嘅工具去研究一維同二維嘅幾何物件。

目錄

Chapter I: Varieties

1. Affine Varieties
2. Projective Varieties
3. Morphisms
4. Rational Maps
5. Nonsingular Varieties
6. Nonsingular Curves
7. Intersections in Projective Space
8. What Is Algebraic Geometry?

Chapter IIː Schemes

1. Sheaves
2. Schemes
3. First Properties of Schemes
4. Separated and Proper Morhpisms
5. Sheaves of Modules
6. Divisors
7. Projective Morphisms
8. Differentials
9. Formal Schemes

Chapter IIIː Cohomology

1. Derived Functors
2. Cohomology of Sheaves
3. Cohomology of a Noetherian Affine Scheme
4. Cech Cohomology
5. The Cohomology of Projective Space
6. Ext Groups and Sheaves
7. The Serre Duality Theorem
8. Higher Direct Images of Scheaves
9. Flat Morphisms
10. Smooth Morphisms
11. The Theorem on Formal Functions
12. The Semicontinuity Theorem

Chapter IVː Curves

1. Riemann-Roch Theorem
2. Hurwitz's Theorem
3. Embeddings in Projective Space
4. Elliptic Curves
5. The Canonical Embedding
6. Classification of Curves in $\mathbb {P} ^{3}$

Chapter Vː Surfaces

1. Geometry on a Surface
2. Ruled Surfaces
3. Monoidal Transformations
4. The Cubic Surface in $\mathbb {P} ^{3}$
5. Birational Transformations
6. Classification of Surfaces

參考書

• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, Zbl 0367.14001
• Shatz, Stephen S. (1979), "Review: Robin Hartshorne, Algebraic geometry", Bull. Amer. Math. Soc. (N.S.), 1 (3): 553–560, doi:10.1090/S0273-0979-1979-14618-4