Geometry Algebraic Geometry by Hartshorne

[micromass]

• Author: Robin Hartshorne
• Title: Algebraic Geometry
• Amazon Link: https://www.amazon.com/dp/1441928073/?tag=pfamazon01-20
• Prerequisities: Commutive algebra, Algebraic Geometry
• Level: Grad

Table of Contents:

[LIST]
[*] Introduction
[*] Varieties
[LIST]
[*] Affine Varieties 
[*] Projective Varieties 
[*] Morphisms 
[*] Rational Maps 
[*] Nonsingular Varieties 
[*] Nonsingular Curves 
[*] Intersections in Projective Space 
[*] What Is Algebraic Geometry? 
[/LIST]
[*] Schemes
[LIST]
[*] Sheaves 
[*] Schemes 
[*] First Properties of Schemes 
[*] Separated and Proper Morphisms 
[*] Sheaves of Modules 
[*] Divisors 
[*] Projective Morphism 
[*] Differentials 
[*] Formal Schemes 
[/LIST]
[*] Cohomology
[LIST]
[*] Derived Functors 
[*] Cohomology of Sheaves 
[*] Cohomology ofa Noetherian Affine Scheme
[*] Cech Cohomology 
[*] The Cohomoiogy of Projective Space 
[*] Ext Groups and Sheaves 
[*] The Serre Duality Theorem 
[*] Higher Direct Images of Sheaves 
[*] Flat Morphisms 
[*] Smooth Morphisms 
[*] The Theorem on Formal Functions 
[*] The Semicontinuity Theorem 
[/LIST]
[*] Curves 
[LIST]
[*] Riemann-Roch Theorem 
[*] Hurwitz's Theorem 
[*] Embeddings in Projective Space 
[*] Elliptic Curves 
[*] The Canonicai Embedding 
[*] Classification of Curves in P^3
[/LIST]
[*] Surfaces
[LIST]
[*] Geometry on a Surface 
[*] Ruled Surfaces 
[*] Monoidal Transformations 
[*] The Cubic Surface in P^3
[*] Birational Transformations 
[*] Classification of Surfaces 
[/LIST]
[*] Appendix: Intersection Theory
[LIST]
[*] Intersection Theory 
[*] Properties of the Chow Ring 
[*] Chern Classes 
[*] The Riemann-Roch Theorem 
[*] Complements and Generalizations 
[/LIST]
[*] Appendix: Transcendental Methods 
[LIST]
[*] The Associated Complex Analytic Space 
[*] Comparison of the Algebraic and Analytic Categories 
[*] When is a Compact Complex Manifold Algebraic? 
[*] Kähler Manifolds 
[*] The Exponential Sequence
[/LIST]
[*] Appendix: The Weil Conjectures
[LIST]
[*] The Zeta Function and the Weil Conjectures 
[*] History of Work on the Weil Conjectures 
[*] The l-adic Cohomology 
[*] Cohomological Interpretation of the Weil Conjectures
[/LIST]
[*] Bibliography 
[*] Results from Algebra 
[*] Glossary of Notations 
[*] Index 
[/LIST]

 

[mathwonk]

This a book developed from courses taught by Hartshorne at Harvard and Berkeley over several years. He taught chapter 4 first on curves, assuming the basic results of cohomology such as riemann Roch, then chapter 5 on surfaces in the same spirit of let's use the big tools on faith.

the reader may be wise to read the chapters in the same order he taught them in, or possibly start with chapter 1, then 4 and 5.

Chapters 2 and 3, on schemes and cohomology are an introduction to the very technical foundations of Grothendieck now universally used in the subject. These chapters are very heavy going, although they are the standard introduction to this dense material.

This is not a beginners introduction to algebraic geometry, for which one can consult books by Miles Reid and Shafarevich. among others. Hartshorne's book however is apparently required reading for grad students in algebraic geometry at top schools, say those wanting to work with professors like Ravi Vakil at Stanford, according to his web page. Be aware that Miles Reid says in his introduction to Shafarevich's more elementary book, that from his own experience many grad students can spend a year or more trying to read Hartshorne with little success. So most of us are advised to have some prior background, say like Reid, Shafarevich, or Miranda's books (plus some commutative algebra and possibly homological algebra).

An introduction to cohomology without the additional complication of schemes is in George Kempf's very concise book Algebraic Varieties. Some of George's arguments are just too brief for me in places, while Hartshorne is usually quite complete.

 

[A. Bahat]

Hartshorne is kind of like a super-condensed version of Grothendieck's EGA. The actual geometry (not that schemes aren't geometry, but you know what I mean--the concrete stuff from which the motivation for the general theory derives) is in Chapters 1, 4, and 5, on Varieties, Curves, and Surfaces, respectively.

It's not a bad book. It's certainly a useful reference. But for learning schemes the first time, I think Ravi Vakil's Stanford notes, "Foundations of Algebraic Geometry", are better. There's more motivation and leisurely explanation in Vakil's notes, and Vakil develops all the algebra he needs (category theory, commutative algebra, homological algebra) from scratch, which really lowers the prerequisites needed, and helps to motivate the algebra too. Hartshorne is a good "road-map" to learning abstract algebraic geometry, but is better off not being read in isolation.

 

[deluks917]

Is there a good reasource on which problems are the most important to do? Some people say just do them all but that is not realistic for most people. There are a lot of problems and many are rather hard.

 

[mathwonk]

just try them all. i.e. do the ones you can do.