Math 818: Algebra and Geometry, Fall 2022

The course syllabus is available here.

Instructor

Dr. Katrina Honigs, khonigs@sfu.ca, Office SC K 10506, Office hours: Mondays 2:30-3:30

Lectures

2:30-4:20 pm WF, AQ 5050

Final Exam Time

Tues. Dec 13. 3:30-6:30 pm, Room K9509

Other links

NTAG seminar schedule

Reading

  • Shafarevich, Basic Algebraic Geometry 1, Available electronically via the SFU Library
  • Fulton, Algebraic Curves, freely available here see also Fulton's Website

    • Problem sets

  • Problem set 1 Due Sept 14
  • Problem set 2 Due Sept 21
  • Problem set 3 Due Oct 5
  • Problem set 4 Due Oct 14
  • Problem set 5 Due Oct 21 Oct 26
  • Problem set 6 Due Nov 2 Nov 4
  • Problem set 7 Due Nov 16
  • Problem set 8 Due Dec 2

    • Lecture schedule:

    Date Subject Book references
    Sept 7 Introduction
    Sept 9 Noetherian rings, Hilbert's Basis Theorem,
    affine algebraic sets    		 Fulton: Ch. 1.1-1.6
    Shafarevich: Ch. I.2.1, Appendix 6
    Sept 14 Ideals of vanishing,
    finitely generated fields, algebras, modules     		Fulton: Ch. 1.7-1.10
    Shafarevich: Appendix 6
    Sept 16 Hilbert's Nullstellensatz,
    Zariski's Lemma     		Fulton: Ch. 1.7-1.10
    Shafarevich: Appendix 6
    Sept 21 Finishing Nullstellensatz,
    irreducibility, coordinate rings     		Fulton: Ch. 2.1
    Shafarevich: I.2.2, I.3.1
    Sept 23 Spec, Zariski topology, localization Ravi Vakil's "The Rising Sea": 3.2-3.5
    Shafarevich: II.1.1
    Sept 28 Gauss's Lemma and irreducibles in A^2
    More with Spec
    Residue fields, generic points, function fields     		Fulton 2.4
    Shafarevich I.3.2
    Sept 30 No Lecture: Truth and Reconciliation Day
    Oct 5 Morphisms, rational maps Fulton 2.2-2.4
    Shafarevich I.3.3
    Oct 7 Intro. to projective space: definitions
    Bezout's Theorem for plane curves (statement)     		Fulton 4.1
    Shafarevich I.4.1
    Oct 12 Homogeneous ideals
    Nullstellensatz for projective space
    Projective space is not affine     		Fulton Ch. 2.6, 4.2, 4.3
    Shafarevich I.4.1, I.4.2
    Vakil 4.4.6-4.4.9
    Oct 14 Morphisms in projective space
    Veronese embedding     		Shafarevich I.4.2, I.4.3, I.4.4
    Oct 19 Classifying quadrics
    Segre embedding     		Shafarevich I.5.1
    Oct 21 Regular maps on projective varieties are closed
    Start of finite maps     		Shafarevich I.5.2, I.5.3
    Oct 26 Finite maps Shafarevich I.5.3
    Oct 28 Dimension Shafarevich I.6.1
    Vakil: 11.1
    Nov 2 Connection between Krull dimension and dimension
    via Noether normalization     		Vakil: 11.2 (particularly 11.2.4)
    Nov 4 Tangent spaces, smoothness
    Bijection between tangent space, Zariski tangent space     		Gathmann's Algebraic Geometry notes, Ch. 10
    Nov 9 Further remarks on tangent spaces
    Dimension of tangent spaces via the Nakayama Lemma     		Shafarevich II.2.2, Appendix A.6 (p. 291/292)
    Nov 11 No Lecture: Remembrance Day
    Nov 16 Discrete valuations
    Weil divisor definition     		Vakil: 12.5, up through 12.5.9
    Silverman's "The Arithmetic of Elliptic Curves" II.3
    Silverman is available online via SFU library access
    Nov 18 Principal divisors
    Examples, elliptic curves     		Silverman II.3
    Nov 23, 25 Kahler differentials, canonical divisors
    Riemann-Roch, genus     		Silverman II.4-5
    Vakil 21.2, up through 21.2.7
    Nov 30, Dec 2 Guest lecturer: Jake Levinson
    Grassmannians     		Gathmann Ch. 8