THEORY OF SCHEMES FALL 2012
PROBLEM SET 1, DUE 9/13
1. Classify, with proof, the points of the ane plane A2 = Spec C[x, y ],
C
and describe the closure of each point in the Zariski topology.
2. Do the same for Spec Z[x] (see Eisenbud-Harris II-37).
3. Do
THEORY OF SCHEMES FALL 2012
PROBLEM SET 5, DUE 10/11
1. Hartshorne II.3.1 locally of nite type.
2. Hartshorne II.3.3 nite type.
3. Hartshorne II.3.6 the function eld of an integral scheme.
4. Hartshorne II.3.13 useful properties of nite type morphisms.
5.
THEORY OF SCHEMES FALL 2012
PROBLEM SET 4, DUE 10/4
1. E-H Exercise II-31 example of a non-at morphism.
2. E-H Exercise II-45 example of a at morphism to Spec Z. (Just the
rst sentence.)
3. Let a1 , . . . , an and b1 , . . . , bn be integers satisfying 0
THEORY OF SCHEMES FALL 2012
PROBLEM SET 3, DUE 9/27
1. Optional, but very strongly recommended.
Access the computer algebra package Macaulay2. Try a computation
that interests you. For example, you can compute the Plcker relations
u
of the Grassmannian of
THEORY OF SCHEMES FALL 2012
PROBLEM SET 2, DUE 9/20
1. Prove that the open subscheme Spec C[x, y ] cfw_(x, y ) of the ane
plane is not itself an ane scheme.
2. Hartshorne II.2.1 distinguished open sets of ane schemes.
3. Hartshorne II.2.3 reduced schemes.
THEORY OF SCHEMES FALL 2012
PROBLEM SET 6, DUE 10/18
1. Hartshorne II.4.1 nite morphisms are proper.
2. Hartshorne II.2.14 parts (a) through (c).
3. E-H Exercises III-12 and III-17 on the Veronese subring.
4. E-H Exercise III-16 closed subschemes and satu
THEORY OF SCHEMES FALL 2012
PROBLEM SET 7, DUE 10/25
Do 4 of the 5 problems.
1. E-H Exercise III-28 projective tangent spaces.
2. E-H Exercise III-29 our two denitions of tangent cones agree.
3. E-H Exercise III-30 examples of tangent cones.
4. Hartshorne
THEORY OF SCHEMES FALL 2012
PROBLEM SET 9, DUE 11/8
1. Vakil 14.1.C through 14.1.H on basic properties of locally free sheaves
2. Vakil 17.4.B on automorphisms of Pn
k
3. Check that the denition of the Veronese map in Vakil 17.4.5 is the
same as the Veron
THEORY OF SCHEMES FALL 2012
PROBLEM SET 12, DUE TUESDAY 12/11 at 5pm
Final version as of Tuesday 12/4.
Please note: no homework will be accepted after Tuesday 12/11
1. E-H IV-69 the limit of the Fano schemes of smooth quadrics degenerating a double plane.
THEORY OF SCHEMES FALL 2012
PROBLEM SET 11, DUE TUESDAY 11/27
1. E-H III-81 a rational quartic curve is not Cohen-Macaulay
2. E-H III-82 example of a failure of Bezout for non-Cohen Macaulay
schemes
3. E-H III-83 reading the degree o of the Hilbert series
THEORY OF SCHEMES FALL 2012
PROBLEM SET 8, DUE 11/1
1. Let be a collection of subsets of cfw_1, . . . , n such that if S and
T S then T . For i = 1, . . . , n, let fi denote the number of sets
in of size i. Let
xj : S cfw_1, . . . , n, S k [x1 , . . . , x
THEORY OF SCHEMES FALL 2012
PROBLEM SET 10, DUE 11/15
1. Vakil 15.2.I Picard group equals class group on a factorial scheme
2. Vakil 15.2.K example of a torsion Picard group
3. Vakil 15.2.Q example of a non Q-Cartier divisor
4. Vakil 15.2.R a criterion fo