# Solutions to Hartshorne's Algebraic Geometry

The goal of this book is to eventually provide a complete, correct, central set of solutions to the exercises in Hartshorne's graduate textbook "Algebraic Geometry". There are many exercises which appear in EGA and a secondary goal would be to have references to all of these.

Please feel welcome to add solutions, correct errors, and add comments on the discussion pages.

This project is very, very young but should grow quickly as latex solutions for many exercises (see below) are available and just need to be transferred here. If you want to help with this process mention it on the discussion page (link top left).

## Contents

edit- Chapter I - Varieties

- Affine Varieties
- Projective Varieties
- Morphisms
- Rational Maps
- Nonsingular Varieties
- Nonsingular Curves
- Intersections in Projective Space
- What Is Algebraic Geometry?

- Chapter II - Schemes

- Sheaves
- Schemes
- First Properties of Schemes
- Separated and Proper Morphisms
- Sheaves of Modules
- Divisors
- Projective Morphisms
- Differentials
- Formal Schemes

- Chapter III - Cohomology

- Derived Functors
- Cohomology of Sheaves
- Cohomology of a Noetherian Affine Scheme
- Cech Cohomology
- The Cohomology 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

- Chapter IV - Curves

- Riemann-Roch Theorem
- Hurwitz's Theorem
- Embeddings in Projective Space
- Elliptic Curves
- The Canonical Embedding
- Classification of Curves in
**P**^1

- Chapter V - Surfaces

## Various solutions available on the web

edit- www.lomont.org/Math/Solutions.pdf
- http://algebraicgeometry.blogspot.com/
- http://www.math.mcgill.ca/bcais/CourseNotes/AlgGeom04/Hartshorne_Solutions.pdf
- http://mathsci.kaist.ac.kr/~jinhyun/sol2/hart.html
- nicf.net/math/rh-IV12.pdf
- http://divisibility.files.wordpress.com/2013/09/fullhartshornesolutions1.pdf

## Solutions which appear in this wikibook

edit- II.1 Sheaves:
- II.2 Schemes:
- II.3 First Properties of Schemes:
- II.4 Separated and Proper Morphisms: 1, 2, 3, 4.
- II.5 Sheaves of Modules:
- II.6 Divisors:
- II.7 Projective Morphisms:
- II.8 Differentials:
- III.2 Cohomology of Sheaves:
- III.3 Cohomology of a Noetherian Affine Scheme:
- III.4 Cech Cohomology:
- III.5 The Cohomology of Projective Space:
- III.6 Ext Groups and Sheaves:
- III.7 The Serre Duality Theorem:
- III.8 Higher Direct Images of Sheaves:
- III.9 Flat Morphisms:

## Solutions which don't appear yet in this wikibook

edit- All of Chapter I
- II.1 Sheaves: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
- II.2 Schemes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19.
- II.3 First Properties of Schemes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.
- II.4 Separated and Proper Morphisms: 5, 6, 8, 9, 10, 11, 12.
- II.5 Sheaves of Modules: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.
- II.6 Divisors: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
- II.7 Projective Morphisms: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.
- II.8 Differentials: 1, 2, 3, 4, 5, 6, 7, 8.
- II.9 Formal Schemes: 1, 2, 3, 4, 5, 6.
- All of Chapter III
- All of Chapter IV
- All of Chapter V

## Solutions to be transferred here for which latex code is available

edit- II.1 Sheaves: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
- II.2 Schemes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19.
- II.3 First Properties of Schemes: 1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 16, 17, 18.
- II.4 Separated and Proper Morphisms: 1, 2, 3, 4, 6, 8, 9.
- II.5 Sheaves of Modules: 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16.
- II.6 Divisors: 1, 4, 11.
- II.7 Projective Morphisms: 1, 2, 4, 5, 6, 7, 8, 9, 12.
- II.8 Differentials: 1, 2, 3, 5, 6, 7, 8.
- III.2 Cohomology of Sheaves: 2, 3, 4, 5, 6.
- III.3 Cohomology of a Noetherian Affine Scheme: 1, 2, 3, 5, 7, 8.
- III.4 Cech Cohomology: 1, 3, 5, 6, 7, 9, 11.
- III.5 The Cohomology of Projective Space: 1, 4, 10.
- III.6 Ext Groups and Sheaves: 1, 3, 4, 5, 6, 7, 8, 9.
- III.7 The Serre Duality Theorem: 3.
- III.8 Higher Direct Images of Sheaves: 1, 2, 3.
- III.9 Flat Morphisms: 1, 2, 7, 9.