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MATH 632: ALGEBRAIC GEOMETRY 2 SCHEME THEORY HOMEWORK ASSIGMENTS PROFESSORKARENE.SMITH Please turn in your problems by the stated dates. By Thursday January 17: Shafarevich Vol 2 Chapter 5 §1: 1, 2, 3, 5, 6 [Correc- tions: In 2, the decomposition is into closed sets, which may not be proper. In 5, take E to be an ideal, not just a set.] By Tuesday January 22: Hartshorne: II §1: 1.2, 1.5, 1.10, 1.12, 1.14. [For these, use Hartshorne’s definition of surjective and injective maps of presheaves.] By Thursday January 24: Hartshorne: II §1: 1.3, 1.6, 1.9, 1.15, 1.17 [Note: Hartshorne’s sheaves are all sheaves of abelian groups.] By Tuesday January 29: Hartshorne: II §1: 1.16, 1.18, 1.21, 1.22 [Note: The notation Hom(F,G) in 18 denotes the set of homomorphism between the named sheaves. This is not the same thing as in problem 1.15, which is a sheaf on X, not a set. Note the different fonts.] By Thursday January 31: Hartshorne II §2: 2.1, 2.2, 2.3, 2.4, 2.7 By Tuesday Feb 5: Hartshorne II §2: 2.8, 2.9, 2.11, 2.12, By Thursday Feb 7: Hartshorne II §2: 2.14, 2.16, 2.18 By Tuesday Feb 12: Hartshorne II §3: 3.1, 3.4, 3.6, 3.7, 3.8 By Thursday Feb 14: Hartshorne II §3: 3.9, 3.10, 3.12, 3.13, 3.21 By Tuesday Feb 19: Hartshorne II §5: 5.1, 5.3, 5.5, 5.6 But do not turn them in to David Speyer; wait for my return and turn them in 2/26. By Thursday Feb 21: Hartshorne II §5: 5.7, 5.8, 5.9 But do not turn them in to David Speyer; wait for my return and turn them in 2/26. By Tuesday Feb 26: Hartshorne II §4: 4.1, 4.3, 4.4, 4.6, By Thursday Feb 28: Hartshorne II §4: 4.5, 4.8 1 2 PROFESSORKARENE.SMITH By Thursday Feb 28: Hartshorne II §4: 4.5, 4.8 BREAK! By Tuesday March 12 : Hartshorne II §5: 5.10, 5.11, 5.12, By Thursday March 14: Hartshorne II §5: 5.13, 5.14, By Tuesday March 19: Hartshorne II §5: 5.17, 5.18. By Thursday March 21: Hartshorne II §6: 6.2ab (note typo in a, the letter i is used in two different contexts), 6.5 abc. Complete QUIZ 4 if you have not already! 4 3 3 4 ByTuesdayMarch26: HartshorneII§7: 7.1,7.2,7.3,ShowthatSpeck[s ,s t,st ,t ] 2 2 2 2 satisfies (∗) but is not normal, and that div (s t ) is effective although s t is not regular. By Thursday April 2: Hartshorne II §7: 7.5, 7.7 By Tuesday April 2: 7.5, 7.7 ByThursdayApril4: Proveallthestatementsregardingthethreedifferentmaps from the blow-up of P2 at a point to different projective spaces discussed in class on March 28. (See the Daily Update.) By Tuesday April 9: Hartshorne III 2.1a, 2.2, 2.7 (Hint: both Q and Q/Z are injective Z-modules.). If you want more, try also 2.3, 2.4. If your homological algebra is weak, study instead Dummit and Foote’s chapter 17, and do some of the exercises there. ByThursday April 11: Hartshorne III 3.1, 3.2, 3.3. If your commutative algebra is strong (eg, you know what “depth” means), you might also like 3.4 and 3.5. By Tuesday April 16: Hartshorne III 4.1, 4.3, 4.7 By Thursday April 18: Hartshorne III 5.1, 5.2, 5.5 [For folks cruising along: 5.3 and 5.6 are interesting and important, and not too difficult if you are following everything.] By Tuesday April 23: Hartshorne IV: 1.1, 1.2, 1.3, 1.4, 1.5, By April 29, for those A+ students who want more: Hartshorne IV: 1.6, 1.7, 3.1. You are now in a pretty good position to read Chapters IV and V of Hartshorne. If you plan to continue in algebraic geometry, this is a good summer plan. [You can go back to II and III as needed, but IV and V are very classical and don’t use most of the fancy stuff from II and III.]
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