Math 818: Algebra and Geometry, Fall 2024

Instructor

Dr. Katrina Honigs, khonigs@sfu.ca
Office SC K 10506
818 Office hours: M 10-11, Th 3-4

Lectures

2:30 2:35-4:20 pm WF, room AQ5051

Final Exam Time

Dec. 12, 2:30-5:30 in AQ 5051

Syllabus

Here

Other links

NTAG seminar schedule

Reading

  • Course notes by Gathmann, particularly drawing from "Plane Algebraic Curves", "Commutative Algebra" and "Algebraic Geometry"

    • Problem sets

  • Homework 0 Due Sept 6, rewrites due Sept 13
  • Homework 1 Due Sept 20, rewrites due Sept 27
  • Homework 2 Due Oct 4, rewrites due Oct 11
  • Homework 3 Due Oct 18, rewrites due Oct 25
  • Homework 4 Due Nov 1 Nov 8, rewrites due Nov 20
  • Homework 5 Due Nov 22

    • Lecture schedule:

    Date Subject Reading
    Part 1: Affine varieties
    Sept 4 Introduction Commutative Algebra Ch. 0-2
    Sept 6 Affine varieties and ideals
    Introduction to modules     		Algebraic Geometry Ch. 1, p. 6-8 (stop before Nullstellensatz)
    Commutative Algebra Ch. 3, p. 26-28
    Sept 11 Noetherian rings
    Hilbert basis theorem     		Commutative Algebra Ch. 7
    We will not cover Artinian rings.
    Sept 13 Irreducibility,
    Zariski's Lemma     		Algebraic Geometry Ch. 1, p. 8-11 (start at Nullstellensatz)
    Commutative Algebra Ch. 10, p. 92-94
    Sept 18 Nullstellensatz Algebraic Geometry Ch. 1, p. 8-11 (start at Nullstellensatz)
    Commutative Algebra Ch. 10, p. 92-94
    Sept 20 Zariski Topology Algebraic Geometry Ch. 2
    Algebraic Geometry Ch. 12, p. 90-92
    Sept 25 Localization Commutative Algebra Ch. 6
    Sept 27 Regular functions Algebraic Geometry Ch. 3 up to 3.13, also 3.19
    Oct 2 Affine morphisms,
    Introduction to projective space     		Algebraic Geometry Ch. 4, p. 32-35,
    Algebraic Geometry Ch. 6
    Part 2: Projective varieties
    Oct 4 Projective space, coordinates
    Start of verifying Bezout for curves     		Algebraic Geometry Ch. 6
    (Optional Plane Algebraic Curves Ch. 2-3)
    Oct 9 Gauss's Lemma
    Projective space and affine cones     		Algebraic Geometry Ch. 6
    Oct 11 Projective Nullstellensatz
    Finish verifying Bezout for curves
    Start of tangent spaces     		Algebraic Geometry Ch. 6
    (Optional Plane Algebraic Curves Ch. 2-3)
    Algebraic Geometry Ch. 10
    Oct 16 Stalks, Zariski cotangent space Algebraic Geometry Ch. 10
    Oct 18 Smoothness, tangent spaces Algebraic Geometry Ch. 10
    Oct 23 Conics and bilinear forms
    Last comments on smoothness, tangent spaces     		Algebraic Geometry Ch. 10
    Oct 25 Projective morphisms Algebraic Geometry Ch. 7
    Oct 30 Projective morphisms are closed
    Grassmannians: Exterior product     		Algebraic Geometry Ch. 7
    Algebraic Geometry Ch. 8
    Nov 1 Grassmannians: Plucker embedding Algebraic Geometry Ch. 8
    Nov 6 Blow-ups Algebraic Geometry Ch. 9
    Dimension and morphisms
    Nov 8 Transcendance degree Commutative Algebra Ch. 11, p. 102-104
    Nov 13 Integral extensions Commutative Algebra Ch. 9
    Nov 15 No class
    Part 3: Riemann--Roch for curves
    Nov 20 Discrete Valuation Rings
    Divisors     		Commutative Algebra Ch. 12 p. 108-11
    Plane Algebraic Curves Ch. 6
    AG notes (older edition) Ch. 14
    Nov 22 Kahler differentials AG notes (older edition) Ch. 15 p. 119-121
    Nov 27
    Nov 29