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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

Reading

  • , 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, rewrites due Nov 29

    • Lecture schedule:

    Date Subject Reading
    Part 1: Affine varieties
    Sept 4 Introduction Ch. 0-2
    Sept 6 Affine varieties and ideals
    Introduction to modules     		, p. 6-8 (stop before Nullstellensatz)
    Ch. 3, p. 26-28
    Sept 11 Noetherian rings
    Hilbert basis theorem     		Ch. 7
    We will not cover Artinian rings.
    Sept 13 Irreducibility,
    Zariski's Lemma     		Ch. 1, p. 8-11 (start at Nullstellensatz)
    Ch. 10, p. 92-94
    Sept 18 Nullstellensatz Ch. 1, p. 8-11 (start at Nullstellensatz)
    Ch. 10, p. 92-94
    Sept 20 Zariski Topology Ch. 2
    Ch. 12, p. 90-92
    Sept 25 Localization Ch. 6
    Sept 27 Regular functions Ch. 3 up to 3.13, also 3.19
    Oct 2 Affine morphisms,
    Introduction to projective space     		Ch. 4, p. 32-35,
    Ch. 6
    Part 2: Projective varieties
    Oct 4 Projective space, coordinates
    Start of verifying Bezout for curves     		Ch. 6
    (Optional Ch. 2-3)
    Oct 9 Gauss's Lemma
    Projective space and affine cones     		Ch. 6
    Oct 11 Projective Nullstellensatz
    Finish verifying Bezout for curves
    Start of tangent spaces     		Ch. 6
    (Optional Ch. 2-3)
    Ch. 10
    Oct 16 Stalks, Zariski cotangent space Ch. 10
    Oct 18 Smoothness, tangent spaces Ch. 10
    Oct 23 Conics and bilinear forms
    Last comments on smoothness, tangent spaces     		Ch. 10
    Oct 25 Projective morphisms Ch. 7
    Oct 30 Projective morphisms are closed
    Grassmannians: Exterior product     		Ch. 7
    Ch. 8
    Nov 1 Grassmannians: Plucker embedding Ch. 8
    Nov 6 Blow-ups Ch. 9
    Dimension and morphisms
    Nov 8 Transcendance degree Ch. 11, p. 102-104
    Nov 13 Integral extensions Ch. 9
    Nov 15 No class
    Part 3: Riemann--Roch for curves
    Nov 20 Discrete Valuation Rings
    Divisors     		Ch. 12, p. 108-11
    Ch. 6
    Ch. 14
    Nov 22 Kahler differentials Ch. 15, p. 119-121
    Nov 27 Riemann--Roch spaces
    statement of Riemann--Roch for curves     		Ch. 8
    Nov 29 Applications of Riemann--Roch to elliptic curves Ch. 7, p. 52-54
    Ch. 8
    For last two weeks of lecture, see also Ch II, III
    of Silverman's "The Arithmetic of Elliptic Curves",
    which is available from the library.