INTRODUCTION TO ALGEBRAIC GEOMETRY
This is the course page for Math 797W: Introduction to algebraic geometry. My office hours this semester will be 12 - 1 Monday/Wednesday/Friday in LRGT 1242.
Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with a strong emphasis on examples. Topics will include projective varieties and schemes, singularities, differential forms, line bundles and sheaves, and sheaf cohomology, including the Riemann--Roch theorem and Serre duality for algebraic curves.
Suggested references
Algebraic Geometry, Hartshorne
The Rising Sea: Foundations of Algebraic Geometry, Vakil
Basic Algebraic Geometry 1 and 2, Shafarevich
Class notes
February 6: Introduction to affine varieties
February 8: Nullstellensatz and correspondence between closed subsets and radical ideals
February 10: Dimension
February 13: Projective varieties
February 15: Regular functions and morphisms
February 17: Introduction to sheaves
No class February 20
February 22: The structure sheaf of projective varieties and rational maps
February 24: Blow-ups and nonsingular varieties
February 27 - March 3: Do all of the Homework 2 Exercises. Class meetings on Zoom.
March 6: Sheaves
March 8: More on sheaves
March 10: Sheaves and Spec
No class March 13 - 17: Spring Break
March 20: Introduction to schemes
March 24: Introduction to schemes, part II
March 27: Introduction to schemes, part III
March 29: Introduction to schemes, part IV
March 31: Introduction to schemes (conclusion) and projective schemes
April 3: Properties and definitions for schemes
April 5: More scheme definitions
April 7: Scheme definitions, with examples
April 10: Fiber products
April 12: Separated morphisms
April 14: Valuative criterion of separatedness
April 19: Proper and projective morphisms
April 21: Projective morphisms
April 24: Introduction to Weil Divisors
April 26: Divisors on affine and projective space
April 28: Examples of class groups
May 1: Cartier divisors
May 3: Cartier divisors and invertible sheaves
May 5: Invertible sheaves and projective morphisms
May 8: (Very) ampleness
May 10: Linear systems and differentials
May 12: Riemann-Roch
Homework
(due to Gradescope every one to two weeks; check this page for due dates)
Homework 1, due February 17
Homework 2, due March 3
Homework 3, due March 24
Homework 4, due April 9
Homework 5, due April 23
Homework 6, due May 19