MAT 5150 — Introduction to Algebraic Geometry
Winter 2009

The syllabus will be updated as the course progresses. Unless otherwise indicated, section numbers and exercises refer to the course text.

Date Sections/Notes Material Recommended Exercises
Jan 7 Introduction Overview of the course, motivation None
Jan 12 A.1 Review of rings Questions in main text of A.1.1, A.1.2; A.4 (Page 207): A.4.1a, A.4.1b, A.4.2a, A.4.2b
Jan 14 A.1 Noetherian rings, factorial rings, basic topology A.4.3a, A.4.3b (Page 208)
Jan 19 I.1 Affine algebraic sets, Zariski topology Page 24: Exercises 1, 2
Jan 21 I.2, I.3 Ideal of an affine algebraic set, irreducibility Page 24: Exercise 3
Jan 26 I.3, I.4 Irreducibility (cont.), Hilbert's Nullstellensatz Page 24: Exercise 4
Jan 28 I.4 Hilbert's Nullstellensatz (cont.) Page 24: Exercise 5
Feb 2 I.5, I.6 Intersection of plane curves, morphisms Page 24: Exercise 6
Feb 4 I.6 Relation between affine algebraic sets and algebras of finite type Page 24: Exercises 7, 8
Feb 9 II.1, II.2, II.3 Projective space Page 34: Exercises 1, 2
Feb 11 II.3, II.4 Conics in projective space, projective algebraic sets Page 35: Exercise 3
Feb 23 II.5, II.7 Graded rings, ideals of projective algebraic sets Page 35: Exercise 4, Page 36: Exercise 5
Feb 25 N/A Midterm Exam N/A
Mar 2 II.6, III.0, III.1 Graded rings associated to projective algebraic sets, sheaves Page 66: Exercise B.1
Mar 4 III.1, III.2 Sheaves, ringed spaces Page 63: Exercise A.1
Mar 9 III.2 Structure sheaf of an affine algebraic set Page 63: Exercise A.2, Page 67: Exercise B.2
Mar 11 III.3, III.4 Affine varieties, algebraic varieties Page 63: A.3a
Mar 16 III.4, III.5 Subvarieties, local rings Page 64: A.4, A.5
Mar 18 III.5, III.6 Local rings, sheaves of modules
Mar 23 III.6, III.8 Sheaves of modules, projective varieties
Mar 25 III.8, III.11 Projective varieties, some more on morphisms Page 64: A.6, Page 68: B.4
Mar 30 IV.1 Dimension
Apr 1 IV.2, V.0, V.1 Dimension, tangent spaces IV.1 (this one will require a bit of work)
Apr 6 V.1 Tangent spaces
Apr 8 V.2, Appendix B Singular points, schemes