Algebraic Geometry (MATH 137)

Algebraic Geometry (MATH 137)

Brooke Ullery | Spring 2020 | Harvard University

This class is an introduction to algebraic geometry. Some topics we will cover include Hilbert's Nullstellensatz, affine and projective varieties, plane curves, Bézout's Theorem, morphisms of varieties, divisors and linear systems on curves, Riemann-Roch Theorem.

General

Lectures

Lecture 1: What is Algebraic Geometry?

Lecture 2: Algebraic Sets

Lecture 3: The Ideal of a Subset of Affine Space

Lecture 4: Irreducibility and the Hilbert Basis Theorem

Lecture 5: Hilbert's Nullstellensatz

Lecture 6: Algebra Detour

Lecture 7: Affine Varieties and Coordinate Rings

Lecture 8: Regular Maps

Lecture 9: Rational Functions and Local Rings

Lecture 10: Affine Plane Curves

Lecture 11: Discrete Valuation Rings and Multiplicities

Lecture 12: Intersection Numbers

Lecture 13: Projective Space

Lecture 14: Projective Algebraic Sets

Lecture 15: Homogeneous Coordinate Rings and Rational Functions

Lecture 16: Affine and Projective Varieties

Lecture 17: Morphism of Projective Varieties

Lecture 18: Projective Plane Curves

Lecture 19: Linear Systems of Curves

Lecture 20: Bézout's Theorem

Lecture 21: Abstract Varieties

Lecture 22: Rational Maps and Dimension

Lecture 23: Rational Maps of Curves

Lecture 24: Blowing Up a Point in the Plane

Assignments

Assignment 1 (PDF) | Assignment 1 (TeX)

Assignment 2 (PDF) | Assignment 2 (TeX)

Assignment 3 (PDF) | Assignment 3 (TeX)

Assignment 4 (PDF) | Assignment 4 (TeX)

Assignment 5 (PDF) | Assignment 5 (TeX)

Assignment 6 (PDF) | Assignment 6 (TeX)

Assignment 7 (PDF) | Assignment 7 (TeX)

Assignment 8 (PDF) | Assignment 8 (TeX)

Assignment 9 (PDF) | Assignment 9 (TeX)

Assignment 10 (PDF) | Assignment 10 (TeX)

Assignment 11 (PDF) | Assignment 11 (TeX)

Texts

Algebraic Curves – Fulton

Algebraic Geometry – Harris

An Invitation to Algebraic Geometry – Smith

Undergraduate Algebraic Geometry – Reid

Basic Algebraic Geometry I – Shafarevich