Math 552 Algebraic Geometry I, Fall 2023

Welcome to Math 552! This course serves as an introduction to Algebraic Geometry. Algebraic Geometry is a central subject in modern mathematics, with close connections with number theory, combinatorics, representation theory, differential and symplectic geometry. We will study basic properties of projective algebraic varieties such as dimension, degree and singularities. At the same time, we will develop a large body of examples that motivate the study of the subject. Depending on time, we will develop the classical theory of curves and surfaces. This course should be enough preparation for a course on the theory of schemes.

Lecturer: Izzet Coskun, icoskun@uic.edu

Drop-in hours: MW 1-2 and by appointment in SEO 423

Venue and time: Lincoln Hall 201, MWF 12-12:50

Text book: There are three recommended texts for this course.

• Joe Harris, Algebraic Geometry: a first course, Springer 1992.
• Igor Shafarevich, Basic Algebraic Geometry I, Varieties in Projective Space, Springer-Verlag 1994.
• David Mumford, Algebraic Geometry I, Complex Projective Varieties, Springer 1995.

Prerequisites: A first year graduate course in algebra: familiarity with commutative rings and modules. We will develop the necessary commutative and homological algebra in the course. However, I strongly recommend taking MATH 531 Advanced Topics in Algebra concurrently. Familiarity with differential geometry or topology helpful, but not required. The material covered in MATH 549 Differentiable Manifolds should nicely complement this course.

Homework: There will be weekly homework. The homework is due on Wednesdays at the beginning of class. Late homework will not be accepted. You may (in fact, you are encouaged to) work on problems togerther; however, the write-up must be your own and should reflect your own understanding of the problem.

Grading: The grade will be entirely based on the homework.

Additional references: The following is a list of references that are more advanced, but you might wish to consult them for more in depth treatments of the subject.

• R. Hartshorne, Algebraic Geometry, Springer 1977.
• D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer-Verlag.
• P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.

Course Notes Available on the Web: The following course notes are really nice. One is more basic and the other more advanced, but you might wish to refer to them.

• Bill Fulton's Notes on Algebraic Curves: Fulton's notes
• Andreas Gathmann's notes on Algebraic Geometry: Gathmann's notes

Lecture notes: I will occassionally post lecture notes here. They are short transcripts of what I planned to say. I make no claims to completeness or accuracy. They are often typed very quickly.

• Notes 1

Course materials: