Spring 2008

TuTh 9:30 - 10:45 LGRT115

**Draft outline of lectures, reading suggestions (mostly from Hartshorne), homework exercises**

* May 8* Kodaira Vanishing Theorem (presented by Jason McGibbon).

* May 6* Birational Invariance of Plurigenera (presented by Amit Datta).
Classification of algebraic varieties.

* May 2* Bertini Theorem (presented by Jason Brunson). Petri Theorem (presented by Adam Gamzon).

* May 1* Linear Systems - 2. Ample linear systems.
Linear systems on curves. Group law on the elliptic curve.

* Apr 24* Normalization. Linear systems.

Exercises:

1. Show that the integral closure of a domain R in a field L is integrally closed.

2 (Jason). Show that finite morphisms are proper using the valuative criterion of properness.

* Apr 22* Normal varieties. III.8 of the Red Book is great, and see also III.9 for
an excellent discussion (much better than in Hartshorne!)
of Zariski's Main Theorem (which we will unfortunately don't have time to cover).

Exercises:

1. Prove that Cl(**P**^{n}_{k}) is isomorphic to **Z**.

2. Suppose that R is a finitely generated
subring of the polynomial ring k[x_{1},...,x_{n}]
of finite codimension over k. Prove that Spec R is not normal but it is normal
outside the finite union of closed points. A cuspidal cubic is a typical example of this set-up.

3. Assuming that R is an integrally closed Noetherian domain,
prove that Cl(Spec R)=0 if and only if R is a UFD.

* Apr 17* Regular local rings are factorial.

Exercises:

1. Show equivalence of three characterizations of projective modules (given in class).

Let M be a projective R-module.

2. Show that any localization of R is projective.

3. If R is local then M is free. Hint: Nakayama.

4. Show that the sheafification of M is a locally free sheaf on Spec R.

Now suppose that M is an R-module such that its sheafification M^{~} is locally free.
Consider a short exact sequence 0->K->N->M->0. The goal of the next set of exercises is to split it.
This will show that M is projective. The argument is a toy version of
the local2global spectral sequence for Ext.

5. Cover Spec R by distinguished affines D_{1},...,D_{n}
such that M^{~} restricted to each D_{i} is free.
Show that one has sections (i.e. splittings)
of localizations f_{i}: M^{~}(D_{i})->N^{~}(D_{i}). Show that the differences f_{i}-f_{j} define a Cech 1-cocycle
for the sheaf Hom(M^{~},K^{~}).

6. Using that Spec R is affine, show that this cocycle
is a coboundary. Use this to tweak f_{i}'s into a splitting M->N.

The goal of the next set of exercises is to show that if (R,m) is a regular local ring
of dimension n with the residue field k=R/m
then Tor^{i}_{R}(k,k)=0 for i>n. Instead of using
the explicit Koszul resolution, we argue by induction on n (this will be a toy version of
the change of base spectral sequence for Tor).
Let x be any element of m\m^{2} and let S=R/(x). Recall that S is a regular local ring
of dimension n-1.

7. Show that Tor^{i}_{R}(R/(x),k)=0 for i>1.

8. Show by induction on p that if M is an S-module such that
Tor^{i}_{S}(M,k)=0 for i>p then Tor^{i}_{R}(M_{R},k)=0 for i>p+1.
Here M_{R} is M viewed as an R-module.

9. Apply the previous exercise to the case k=M to conclude that k (and therefore any R-module)
has a free resolution of length at most n.

* Apr 15* Cartier divisors vs. Weil divisors on non-singular varieties.

Exercises:

1. Show that a ruling of a quadratic cone xy=z^{2}
is not locally given by one equation.

2. Show that if R is a UFD then any prime ideal of codimension 1
is principal.

* Apr 10* Conormal sequence for smooth subvarieties. Picard group and Cartier divisors.

Exercises:

1. Show that any invertible sheaf (on an integral scheme)
is isomorphic to a subsheaf of the constant sheaf of rational functions.

2. Show that Pic(X) is isomorphic to the group of Cartier divisors modulo principal Cartier divisors.

* Apr 8* Conormal sequence and adjunction.

Exercises:

1. Suppose that 0->F_{1}->F_{2}->F_{3}->0
is an exact sequence of coherent sheaves.
Show that if F_{2} and F_{3} are locally free then F_{1} is locally free.
Are the other two similar-looking statements true?

2. Show that a smooth quartic surface in P^{3}, a smooth complete intersection
of a quadric and a cubic hypersurfaces in P^{4}, and
a smooth complete intersection of three quadrics in P^{5} are K3 surfaces.

* Apr 3* Pull-back and push-forward of quasi-coherent sheaves.
The lecture was based on
this handout of Vakil.

Exercises:

1. Prove that pull-back and push-forward are adjoint for affine schemes,
i.e. that Hom_{A}(A⊗_{B}N,M)=Hom_{B}(N,M_{B})

2. Let X be a scheme, x its point, and consider the natural morphism f:Spec k(x)->X.
Show that f^{*}F is nothing but the fiber of F at x.

3. Let f:X->Y be a morphism. What is the pull-back of O_{Y}?
Show that the pull-back of a coherent sheaf is coherent.

4. Let f:X->Y be a closed embedding and let F be a quasicoherent sheaf on X.
Show that f^{*}(f_{*}(F)) is isomorphic to F.

* Apr 1* Nonsingularity = Cotangent sheaf is locally free.

Exercises:

1. Prove the relative cotangent sequence: if R->S->T are maps of rings, then there is
a right-exact sequence of T-modules
T⊗_{S}Ω_{S/R} ->Ω_{T/R}->Ω_{T/S}->0.

2. Check that the map B⊗_{A}B -> B⊕Ω_{B/A},
given by b⊗b'->(bb',bdb'), is a ring homomorphism

* Mar 27* Sheaf of Kahler differentials. Its calculation for the projective space.
Zariski cotangent spaces as fibers of this sheaf at closed points.

Exercises:

1. Check the universal property of the module of Kahler differentials.

2. Deduce from the Euler exact sequence that the canonical sheaf
of P^{n} is isomorphic to O(-n-1)

* Mar 25* Zariski (co)tangent spaces, nonsingular varieties, Jacobian criterion,
regular local rings, Nakayama's lemma. Next time we will introduce the sheaf of Kahler differentials.
Reading: Red Book III.4 (the definition of Kahler differentials is hidden in III.1).
Hartshorne II.8

Exercises:

1. Check that schemes Spec k[x,y]/(xy) and Spec Z[2i] are not regular.

2. Suppose that algebraic varieties X and Y are non-singular at closed points x and y, respectively.
Prove that XxY is non-singular at (x,y).

* Mar 13* We discussed Serre's theorems on cohomology of coherent sheaves F on P^{n}_{R}:
global generation of F(m) for m>>0,
H^{i}(F) is a finitely generated R-module,
H^{i}(F(m))=0 for i>0, m>>0,
and Serre's correspondence. Roughly this corresponds to III.5 of Hartshorne.

Exercises:

1. O(m) of P^{n}_{R} is globally generated if and only if m>=0.
If F(m) is globally generated then F(n) is globally generated for n>m.

2. Suppose M and N are R-modules and F, G are associated sheaves on Spec R.
Show that the tensor product of F and G is associated with a tensor product of M and N
(Hint: tensor product of modules commutes with localization).

3. Show that O(m) tensor O(n) is O(m+n).

4. Show that the global generation theorem is a formal consequence of Serre's correspondence.

The goal of the next exercises is to show that Serre's correspondence
follows from the global generation theorem.
For any sheaf F, consider the graded S-module M(F): the direct sum of
global sections of F(m) for all positive m. Let S(F) be its sheafification.
Serre's correspondence asserts that F is isomorphic to S(F).

5. Let M(F)^{k} be the truncation of M(F): it is equal to M(F)
in degrees at least k and in degrees smaller than k it is equal to 0. Show that the sheafification
of M(F)^{k} is isomorphic to S(F).

6. Show that a short exact sequence of sheaves 0->F_{1}->F_{2}->F_{3}->0
induces a short exact sequence of sheaves
0->S(F_{1})->S(F_{2})->S(F_{3})->0
(Hint: Serre's vanishing and the previous exercise).

7. Construct a `natural' map of sheaves S(F)->F.

8. Show that S(F) is isomorphic to F if F is a direct sum of twisting sheaves
and then use global generation to show that S(F)->F is surjective.
Then prove it is injective!

9. It remains to prove that M(F) is finitely generated.
Prove first that graded components are finitely generated R-modules.

10. Consider the map from F(m)+...F(m) (n+1 times) to F(m+1)
given by multiplication of sections with x_{0},...,x_{n} and adding.
Use global generation theorem (and Serre's vanishing) to show that this map
is surjective for m>>0. Conclude that M(F) is a finitely generated S-module.

* Mar 11* We finished the calculation of cohomology of O(m) on the projective space.
We also introduced coherent sheaves (on Noetherian schemes)

Exercises:

1. Show that if F is a coherent sheaf on X and U=Spec R is its affine open then H^{0}(U,F)
is a finitely generated R-module.

2. Show that if Y is a closed subscheme of X then the pushforward of a coherent sheaf on Y
is a coherent sheaf on X.

3. Let S=R[x_{0},...,x_{n}] be a ring of polynomials
and let I in S be a homogeneous ideal (not containing S_{+}).
Let Y=Proj(S/I). Let F be a quasicoherent sheaf on Y and let O_{Y}(m) be the twisting sheaf of Serre.
Let F(m) be F tensored with O_{Y}(m). Note that Y is a closed subscheme of P^{n}_{R}.
Prove that the pushforward of F(m) is isomorphic to the pushforward F tensored
with O_{PnR}(m)

4. Show that Coh(X) is an Abelian subcategory of QCox(X).

* Mar 6* Cohomology of projective schemes. Vanishing of cohomology in degrees higher than dimension.
We also discussed dimension of schemes. Cohomology of O(m) on the projective space.

Exercises:

1. Show that any irreducible closed subset of a scheme has a unique generic point.

2. Show that any closed subset of a Noetherian scheme contains a closed point.

3. Let S be a graded ring and I in S a homogeneous ideal (not containing S_{+}). Show that
Proj S/I has a canonical closed immersion in Proj S such that its ideal sheaf is a sheafification of I.

4. Prove 'Prime avoidance': suppose I_{1}, ..., I_{l} are prime ideals of R
and J is any ideal.
If J is not contained in any I_{i} then J is not contained in the union of I_{i}'s. Show that if everything is graded then one can find a homogeneous element of J
not contained in the union of I_{i}'s.

* Mar 4* Cech cohomology of quasi-coherent sheaves. Reading: Vakil's handouts
29
and
30.

Exercises:

1. Check that d^{2}=0 in the Cech complex.

2. Show that refinement of covers induces maps of corresponding Cech complexes.

3. Exercise III.4.7 in Hartshorne.

* Feb 28* Separated and proper morphisms-II. Reading: Hartshorne II.4 until the end
(skip proofs of valuative criteria)

Exercises:

1. Finish the proof of the theorem that characterizes the functor of points of the projective space.

2. Show that the functor of global sections on the affine scheme is exact.

3. Check as many statements as possible in Corollary 4.8 using the valuative criterion.

* Feb 26* Separated and proper morphisms. Reading: Hartshorne II.4 until 4.8 (skip proofs of valuative criteria)

Exercises:

1. Prove the functorial characterization of a reduced induced subscheme.

2. Finish the proof that closed immersions are stable under base change.

3. Prove that the affine line with the origin doubled is not separated.

4. Check as many statements as possible in Corollary 4.6 using the valuative criterion.

* Feb 21* Functor of points. Fiber product.

* Feb 14* Morphisms of schemes. Closed embeddings.

Exercises:

1. Show that there is a natural bijection between morphisms X->Spec R and homomorphisms
R->H^{0}(X,O_{X}) assuming this is known if X=Spec S.

2. Show that if S is a graded algebra generated over S_{0} by r+1 elements of S_{1}
then there is a natural closed immersion Proj S->P^{r}_{S0}.

3. Show that the category of quasicoherent sheaves is Abelian, i.e. closed
under taking kernels, cokernels, and finite sums.

4. Show that there is a natural bijection between closed subschemes of X and quasicoherent subsheaves
of ideals of O_{X}.

5. Prove that any closed immersion X->Spec R corresponds to a surjection of rings R->R/I
(Hint: define I as global sections of the sheaf of ideals of X).

* Feb 12* Proj. Quasicoherent sheaves on Proj. Reading: Hartshorne II.2 (Proj), II.5 (sheaves on Proj)

Exercises:

1. Check that subsets V(I) of Proj R satisfy axioms of closed subsets of a topology.

2. Prove that D_{+}(f) intersects D_{+}(g) in D_{+}(fg)
("convexity" of distinguished opens) and that D_{+}(f) is contained in D_{+}(g)
if and only if f^{n}=gh for some n and h.

3. Let S be a graded ring, let f be an element of S_{1} and let g be an element of S_{r}.
Suppose D_{+}(g) is contained in D_{+}(f).
Then identify D_{+}(g) with a distinguished affine open D(g/f^{r}) of Spec (R_{f})_{0}
Now let M be a graded S-module. Show that (M_{g})_{0} is isomorphic to
((M_{f})_{0})_{g/fr}.

4. Let R be a graded ring and let S=R_{0}+R_{d}+R_{2d}+R_{3d}+...
be its *Veronese subring*. Prove that Proj R=Proj S.

* Feb 7* "Affine local properties".
Reading: Section 3 of this handout
and Section 1 of this handout.

Exercises:

1. A scheme X is called a scheme of locally finite type over a field K if it has an affine cover by
affine open sets U_{i} such that each section ring O_{X}(U_{i})
is a finitely generated K-algebra. Prove that in this case
O_{X}(U) is a finitely generated K-algebra for any affine open set U.

2. More generally, given a morphism of schemes f:X->Spec R,
X is called a scheme of locally finite type over Spec R (or over R)
if it has an affine cover by affine open sets U_{i} such that
each section ring O_{X}(U_{i})
is a finitely generated R-algebra
(explain why is it an R-algebra in the first place). Prove that in this case
O_{X}(U) is a finitely generated R-algebra for any affine open set U.

3. Even more generally, a morphism of schemes f:X->Y is called locally of finite type
if Y has an affine cover by affine open sets V_{i} such that
each f^{-1}(V_{i}) is a scheme of finite type over V_{i}
(in the sense of the previous definition).
Prove that in this case
f^{-1}(V) is a scheme of finite type over V
(in the sense of the previous definition) for any affine open set V of Y.

4. Prove that the morphism of schemes f:X->Y is locally of finite type if and only if for any
affine open U of Y and for any affine open V of f^{-1}(U), O_{Y}(V) is a finitely generated
O_{X}(U)-algebra.

5. Fun exercise from the VGS lecture of Harm Derksen: let R be a k-algebra and let B be the set
of elements f in R such that R_{f} is a finitely generated k-algebra. Prove that B is an ideal!
A fun part is to show that if R_{f} and R_{g} are finitely generated then R_{f+g}
is finitely generated. Hint: reinterpret everything in terms of schemes of finite type.

6. A scheme X is called locally Noetherian if
it has an affine cover by affine open sets U_{i} such that each section ring O_{X}(U_{i})
is a Noetherian ring. Prove that in this case
O_{X}(U) is a Noetherian ring for any affine open set U.

*Feb 5*: Algebraic varieties vs. Algebraic schemes.
Quasicompact, irreducible, reduced, integral schemes. Schemes of finite type over a field,
Suggested Reading: II.3 of Mumford.

Exercises:

1. An affine scheme is quasicompact.

2. A scheme is quasicompact if and only if it admits a finite cover
by affine open subsets.

3. A scheme is reduced if and only if all stalks of O_{X}
have no nilpotents.

4. A scheme is reduced if and only if
it admits a covering by affine opens U_{i}
such that O_{X}(U_{i}) has no nilpotents for any i.

5. A scheme is integral if and only if O_{X}(U) is a domain for any open U.

*Jan 31*: Fun with quasicoherent sheaves :)

Exercises:

1. The residue field at the prime ideal P in Spec R is equal to the quotient field of R/P.

2. Localization is an exact functor. Also, if M->M'->M'' is an exact sequence of R-modules
then the correspondent sequence of associated quasicoherent sheaves is exact as well.

3. Finish the proof of the existence theorem of an associated quasicoherent sheaf:
show that a section over a distinguished open D(f) is uniquely determined by (and can be glued from)
its restrictions to some open cover of distinguished opens.

4. A sheaf F of O_{X} modules is quasicoherent if and only if
locally there exists an exact sequence of O_{X} modules
O_{X}^{I}->O_{X}^{J}->F->0.
Here I and J are some indexing sets, not necessarily finite.

*Jan 29*: Locally ringed spaces, sheaves of O-modules. Affine schemes. Schemes. Quasicoherent sheaves.

Reading: II.1 (sheaves), II.2 (schemes), II.5 (until 5.8, sheaves of modules, quasicoherent sheaves).
We will keep working on these topics next time.

Exercises (thanks Jason for pointing out an error in #6)

1. Zariski topology on Spec R is a topology.

2. Distinguished open sets D(f) form a base of Zariski topology on Spec R.

3. Intersection of two distinguished opens is a distinguished open.

4. The radical of an ideal I is equal to the intersection of prime ideals containing it.

5. When is V(I) equal to V(J)?

6. Show that D(f) is contained in D(g) if and only if some power of g divides f (in R).
** Course policies**

This is the second part of the course in Algebraic Geometry.
We will concentrate on the language of schemes and on the tools
(theorems and basic principles used by geometers).
If you don't know what a scheme is, don't worry: we will define it in the first class!
The main textbook will be Hartshorne's *Algebraic Geometry*
but looking in Mumford's *Red Book* will often help as well.
People proficient in French could find
Grothendieck's EGA surprisingly easy to read.
Another web resource that I find very useful is a
set of lectures by Ravi Vakil.

The minimal set of prerequisites is a good knowledge of Sections 1.1-1.6 and 2.1 of Hartshorne
including understanding of concepts and results from Commutative Algebra quoted there.
However, ideally you should also be familiar with Sections 1.1-1.9 of the Red Book
(covered in Eyal Markman's class).

This semester we will try a risky experiment of synchronizing the course with the
reading seminar in Algebraic Geometry.
This informal seminar will be accessible to graduate students and an active participation will be (actively) encouraged.
The goal will be to discuss how the tools of Algebraic Geometry are applied in practice,
using as an example the recent progress in solving Green's conjecture by Claire Voisin.
The secondary goal will be to gain more experience with Commutative Algebra.
There will be a lot of interaction between the course and the reading seminar,
so students who would not be able to participate in this reading group should let me know as
soon as possible to arrange for an independent research project.

The homework problems will be posted on this web-site. Most of the time these problems
will be `leftovers' from the proofs given in the class. My expectation is that
each student will talk to me as often as possible (but at least once in two weeks)
about his or her progress in doing these exercises. I also expect that you will honor
my requests to meet more often if I find this necessary.
You will pass this course based on your active participation in the class and in the reading seminar,
and on your efforts in solving homework problems.

This semester I will have `open doors' policy: you are wellcome to come to my office 1236
anytime to talk about Algebraic Geometry. If you want to be sure that I am around,
stop by on Monday or Wednesday 3-4:30 and on Thursday 2:30-3:30.