Math 511, Algebraic Geometry I, Spring 2005

• Review of affine varieties
• Basics of sheaves
• Basics of affine schemes
• Basic examples of affine schemes
• The dimension of a scheme
• First properties of schemes
• Fiber products
• Sheaves of modules
• Midterm and Solutions
• Projective varieties
• Bezout's Theorem
• Projective schemes
• Ample sheaves
• Separated and proper morphisms
• DVRs, regular rings, and Kahler differentials
• Divisors, part I
• Divisors, part II

• Week 1: Read Hartshorne, Chapter I, sections 1, 3, 4. In section 1, do problems 1.1(a,b), 1.2, 1.3, 1.4, 1.6; section 3, problems 3.2, 3.15; section 4, problem 4.1, all to be turned in on Friday, January 28.
• Week 2: Read Hartshorne, Chapter II, section 1. In section II.1, do the following problems, due Friday, February 4: essential (to be turned in): 1.1-1.8, 1.13, 1.17; important (but need not be turned in):1.9-1.12, 1.14, 1.21; optional (but useful!): 1.15, 1.16, 1.18-1.20.
• Week 3: Read Hartshorne, Chapter II, section 2. Do the following problems for Wednesday, February 16.
• Week 5+: Read Hartshorne, Chapter II, section 3. Also read Chapter II, section 5. In Section 3, do problems 1, 4, 6, 9, 10, 16 to turn in on Wednesday, March 2. Also, do problems 2, 3, 5, 7, 8, 11, 13, 14 for your own education (not to turn in).
• Week 7: Read Hartshorne, Chapter II, sections 5 and 4. Do problems 5.1, 5.3, 5.7, 5.8, 5.17, 5.18 for Friday, March 11--UPDATE: postponed--due date Monday, March 14.
• Week n: Read Hartshorne, Chapter I, section 2, and Chapter II, sections 2, 4, and 5. Do the homework n for Wednesday, April 6.
• Week n+1: Read Hartshorne, Chapter II section 4. Do problems I.2.15, I.2.16, II.4.1, II.4.2, II.4.7, to be turned in on Friday, April 15. Also do II.4.8, II.4.9, but do not turn them in.
• Week n+2: Read Hartshorne, Chapter I, sections 4, 5, 6. Do problems I.6.1, I.6.2, and I.6.4, to be turned in on Monday, April 25.
• Week ∞: Read Hartshorne, Chapter I, section 7. Read Chapter II, sections 6,7,8. Do problems II.6.1, I.7.2(a,b,c,e) (this should be easier if you imitate our proof of Bezout's theorem to compute Hilbert series); II.8.3. Also, determine the divisor of x/y on the quartic surface xy-zt=0 in P³. These problems need not be turned in.