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.]