Algebraic Geometry

Professor A.J. de Jong, Columbia university, Department of Mathematics.

The plan of this semester course in algebraic geometry is to start developing the basic theory of schemes.

We will use the book [H] = Hartshorne on algebraic geometry. Most of the material can also be found in the stacks project.

It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday 8:40-9:55 in Math 507.

Problem sets will be announced on this web page after Tuesday's lecture. They are due in lecture on the next Tuesday.

The TA for the course is Natasha Potashnik.

Grades are computed by a weighted average between the scores on problem sets and final. The weights are 2/3 and 1/3 respectively.

The final will be a written exam.

Here are the weekly problem sets. Please hit the refresh button on your browser to make sure you have to latest list. These exercises are partially meant for you to see if you know enough to be able to follow the material in the course. Hence it is suggested that you skip the ones you are familiar with, or give a very brief answer showing you understand the point. Moreover, most of the exercises are of a theoretical nature, hence you'll be able to look up that answer -- feel free to do this.

• First problem set due January 21.
  1. Read in the book Algebraic Geometry by Hartshorne, Chapter II, Sections 1, 2, 3, 4.
• Second problem set due January 28.
  1. Let f : X ---> Y be a morphism of schemes. We say f is quasi-compact if the inverse image of every quasi-compact open is quasi-compact. Prove that f is quasi-compact if and only if there is a covering of Y by affine opens V_j such that f^{-1}(V_j) has a finite covering by affine opens of X.
  2. Give a detailed proof of Lemma Tag 00EQ (this is to exercise your ``glueing muscle''; you can replace this by a detailed proof of another glueing lemma of this type if you like).
  3. Read about Yoneda's lemma (for example Section Tag 001L) and about fibre products of schemes (Chapter II, Theorem 3.3 in [H]).
  4. Write out completely why "locally of finite type" is preserved under base change.
  5. Write out completely why "being surjective" is preserved under base change.
• Third problem set due Feb 4.
  1. Let k be a field. Let k[X_0, ..., X_n] be the graded k-algebra where each X_i is homogeneous of degree d_i (not necessarily equal to 1). Let P = Proj(k[X_0, ..., X_n]) viewed as a scheme over k, i.e., over Spec(k). Roughly speaking P should be viewed as (k^{n + 1} - {0})/k^* where the action of t ∈ k^* is given by multiplication by t^{d_i} on the ith coordinate. Let us find out to what extend this is true by studying points of P over k. Recall that a k-rational point of a scheme X over k is either a point x ∈ X whose residue field is k (via the given map from k into it) or equivalently a morphism Spec(k) ---> X such that the composition Spec(k) ---> X ---> Spec(k) is the identity. The set of k-rational points is denoted X(k).
     1. Show that there is a way to associate to an element of k^{n + 1} - {0} a k-rational point of P.
     2. Show that the map k^{n + 1} - {0} ---> P(k) factors through the action of k^* described above.
     3. Show the map is surjective.
     4. Show that (k^{n + 1} - {0})/k^* ---> P(k) is not injective in general.
  2. Let k be a field. Let X be a locally closed subscheme of P^n_k. Prove that X is not closed in P^n_k if and only if there exists a curve C (a dimension 1 reduced and irreducible scheme) which is closed in X but not closed in P^n_k. (Hint: use induction on the dimension of X and for that use everything we know about dimensions from last semester.)
  3. For which schemes X is every morphism of the form Spec(A) ---> X an affine morphism and why?
  4. Lefschetz principle: Let X be a projective scheme (variety) over the complex numbers C. Show there exists a finite type Z-subalgebra A ⊂ C and a projective morphism Y ---> Spec(A) such that X is the base change of Y via the morphism Spec(C) ---> Spec(A).
• Fourth problem set due Feb 11.
  1. Hand in the solutions to the previous problem sets.
  2. Read in Hartshorne all the way through the first part of II, Section 5.
• Fifth problem set due Feb 18.
  1. Read Hartshorne, Chapter II, Sections 1, 2, 3, 4, 5.
  2. Give an example of a scheme X and a quasi-coherent sheaf F on X which is not locally free.
  3. Give an example of a scheme X projective over a field k which is not isomorphic to a closed subscheme of P_k^2.
  4. Let X be a projective scheme over a ring A. Show that every finite set of points of X is contained in an affine open of X.
  5. Give an example of a scheme X and a locally free sheaf E on X which is not isomorphic to a direct sum of invertible sheaves. Discussion: Even though almost every scheme X has such a vector bundle, I don't know any really easy examples. I'm hoping you'll find one. You can try reformulating this into algebra (for affine X) and solving the corresponding algebra problem. Or you can try taking P^2 and some vector bundle on P^2. Another option is to take X equal to affine three space with coordinates x, y, z with the origin 0 = (0, 0, 0) removed. Then consider the surjective map O_X^3 ---> O_X given by (x, y, z) and let E be the kernel. Show that this E is not a direct sum of invertible sheaves using that every invertible sheaf on X is trivial (also not easy) and the algebraic version of Hartog's theorem.
• Sixth problem set due March 4
  1. Keep up by reading Chapter II, Section 6 in Hartshorne.
  2. Also, please look at the very instructive exercises in Chapter II, Section 6 of Hartshorne, especially the one about cones (II.6.3) and the one about quadric hypersurfaces (II.6.5). Try to think through some of the steps and (optionally) write it out and hand it in to be graded.
  3. Let C, D be irreducible curves in P^2 (over an algebraically closed field) of degrees a, b. Let U be the complement of their union in P^2. What is Cl(U)?
  4. Let A ---> C and B ---> C be surjective ring maps with kernels I and J, so C = A/I = B/J. Let R = A x_C B be the fibre product of rings, i.e., R = {(f, g) ∈ A x B | f mod I = g mod J}.
     1. Show that φ : R --> A and ψ : R --> B are surjective ring maps with kernels J and I respectively.
     2. Show that Spec(R) is the union of closed subsets homeomorphic to Spec(A) and Spec(B) glued along the common closed subset Spec(C)
     3. Show that if f ∈ R then R_f = A_{φ(f)} x_{C_g} B_{ψ(f)} where g ∈ C is the image of f.
     4. Show that if a point x of Spec(R) is contained in Spec(A) and not in Spec(C), then there is an f ∈ R such that R_f = A_{φ(f)}.
     Next, let M be a finite locally free A-module and let N be a finite locally free B-module. Let α : M/IM ---> N/IN be an isomorphism of C-modules. Set P = {(x, y) ∈ M x N | α(x) = y} = M x_{M/IM = N/JN} N.
     1. Prove the analogue of c above for P and M and N.
     2. Show that P is a finite locally free R-module. Hint: By localizing, using part a we see that we may assume M and N are free. Then α is given by an invertible matrix. After shrinking the open neighbourhood, we may assume that α lifts to an invertible matrix with coefficients in A or B. On such a neighbourhood the module P becomes free.
     3. Using the above, show that there exists an exact sequence A^* x B^* ---> C^* ---> Pic(R) ---> Pic(A) x Pic(B) ---> Pic(C)
     4. Let X be the banana curve: glue two copies of the affine line over a field k in two distinct points. In other words, say A = k[t], B = k[t], and C = k x k where the maps φ and ψ are given by evaluating at 0, 1, namely f |---> (f(0), f(1)). Then set X = Spec(R). Using the above show that Pic(X) = k^*.
• Seventh problem set due April 1 (not an April fool's joke).
  1. Keep up by reading Hartshorne, Chapter II, Sections 7 and 8.
  2. Let b : X ---> P^2 be the blowing up of a closed point in P^2 over an algebraically closed field k. Show that there is a morphism X ---> P^1 which induces an isomorphism of the exceptional fibre of b with P^1.
  3. Let X, Y be projective varieties over an algebraically closed field k. Let U ⊂ X be a nonempty open subset and let f : U ---> Y be a morphism. Show there exists a blow up b : X' ---> X whose center is disjoint from U such that f extends to a morphism of varieties X' ---> Y.
  4. For which values of the parameter t is the projective surface in P^2 with equation X_0^4 + X_1^4 + X_2^4 + X_3^4 + tX_0X_1X_2X_3 = 0 singular?
  5. Show that the strict transform of the plane curve with equation xy(x - y) + x^7 + y^7 = 0 under blowing up in (0, 0) is nonsingular.