MATH 818: Algebra and Geometry

Instructor: Nils Bruin
SC K 10507
(778) 782 3794
Outline: Algebraic geometry is a topic central to pure mathematics, and its ideas and results play an ever increasing role in all fields of mathematics. This makes it a very worthwhile topic to learn about if you are interested in mathematics.

This course provides an introduction into algebraic geometry at the graduate level with an emphasis on the one-dimensional case: the geometry of algebraic curves.

The exact content of the course will be adapted to those participating in it and will be determined during the first meeting, Wednesday September 7, 2016.

Topics will likely include:

  • basic notions of affine and projective varieties
  • the notions of rational maps between varieties
  • differentials and singularities
  • function fields
  • Divisors, the Riemann-Roch theorem
  • classification of algebraic curves

Course text: Shafarevich, I. R. Basic algebraic geometry, part 1: Varieties in Projective Space; Springer-Verlag (1977) ISBN 0-387-54812-2

Recommended reading:
Lectures: First Lecture: September 7, 14:30 - 16:20 in AQ 5006
Regular Lectures: Wednesday, Friday, 14:30 - 16:20 in AQ 5006 (note some cancelled dates!)
Replacement Lectures: Monday, 14:30 - 16:20 in SECB 1010 (on select dates)
Exam: Preliminary schedule:
Take-home exam on Tuesday, December 6, 2016; 9am - 5pm
Assignments: Biweekly, posted here.
Preliminary scheme:
Assignments: 30 %
Presentation: 20 %
Exam: 50 %
Lecture schedule: BOLDFACE DATES are Monday lectures.
Date Subject Book references
Sep. 7 Planning and Introduction
Sep. 9 Affine algebraic sets, Hilbert's Basis Theorem Shafarevich: Ch. I.2.1, Appendix 6
Fulton: Ch. 1
Sep. 14 Hilbert's Nullstellensatz Shafarevich: Appendix 6; 3.1
Fulton: Ch. 1
Sep. 16 Coordinate rings, irreducible algebraic sets, morphisms Shafarevich: I.2.2, I.2.3, I.3.1
Fulton: 2.1-2.3
Sep. 21 Function fields and Birational geometry Shafarevich I.3.2, I.3.3
Fulton: 2.1-2.4
Sep. 23 Projective space and their closed sets Shafarevich: 4.1, 4.2
Fulton: Chapter 4
Sep. 26 Projective coordinate rings, Quasiprojective varieties Shafarevich 4.2
Sep. 28 Quasiprojective varieties: rational maps, regular maps on projective varieties are closed Shafarevich I.4.3, I.4.4, I.5.2, I.6.1
Oct. 5 Finite Maps; Noether Normalization; Dimension Shafarevich I.5.3, I.5.4, I.6.1
Oct. 7 Dimension of Projective (sub)varieties; Dimension of Fibres Shafarevich I.6.2, I.6.3
Oct. 12 Singular and nonsingular points Shafarevich II.1.1-3
Oct. 14 Local parameters and series expansions; discrete valuation rings Shafarevich II.2.1-2
Fulton 2.4,2.5
Oct. 17 Discrete Valuation Rings; Multiplicity of singular points Fulton 3.2
Oct. 19 Blow-up, normalization Fulton 7.2, Shafarevich II.5.1
Oct. 21 Normal varieties; divisors on curves Shafarevich II.5.1, III.2.1
Oct. 24 Pull-back of divisors; ramification degrees Shafarevich III.2.1
Oct. 26 Bezout following Fulton Fulton 5.3
Oct. 28 Dimension of Riemann-Roch spaces Fulton 8.1, 8.2
Nov. 7 Riemann's Theorem Fulton 8.3
Nov. 9 Derivations and Kähler differentials Fulton 8.4, any reference on Kähler differentials
Nov. 11 REMEMBRANCE DAY university closed
Nov. 14 Differentials and the canonical class Fulton 8.5
Nov. 16 Riemann-Roch for smooth plane curves Fulton 8.6
Nov. 18 Classification of genus 1 curves and their automorphism groups Hartshorne IV.4
Nov. 23 Further topics: Schemes and group varieties
Nov. 28 Presentations
Nov. 30 Presentations
Dec. 6 EXAM