Algebraic Geometry MATH GR6262

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

This semester I am teaching the course on schemes. If you are interested, please email me, tell me a little about yourself (academically), and I will add you to my email list.

The first lecture will be Wed, Jan 22 at 4:10pm - 5:25pm in 507 Math. The schedule will be Monday and Wednesday at 4:10 in 507 Math.

To keep up with the course it is **very important** to both attend the lectures and to work on the problem sets. I know that my problem sets are often impossible and grading will reflect this. Please focus on doing some of the problem set each week and do not try to do all of the problems completely.

There is no TA. I will have office hours starting at 3:00 pm on Thursday in my office and continuing through tea (I will be in the common room during tea time).

Exam. Thu, May 12 2025 at 4:10-7:00pm in MATH 507

Content. The goal is to esthablish the existence of the Picard variety or Jacobian of a (projective, smooth) curve and to deduce some properties of the Picard group of a curve from this (using a tiny bit of the geometry of abelian varieties). This is too much to cover in a 1 semester course, so we will have to blackbox lots of material. Still, I think we can carefully work through the key arguments of the proof.

Readings. We will use the exposition in the Stacks project, especially the chapter on Picard Schemes of Curves and the section on Abelian varieties.

Here is a preliminary list of things I intend to discuss:

• Schemes, (some) properties of schemes, morphisms of schemes, (some) properties of morphisms of schemes, quasi-coherent modules, (some) properties of quasi-coherent modules, Picard groups.
• Glueing lemma, Yoneda lemma, characterization of representable functors, existence of the fibre product as an example.
• Separation axioms, properness, projective schemes.
• Weil divisors, effective Cartier divisors, rational equivalence, and Picard groups.
• Smoothness and nonsingularity.
• Hilbert functor of points, in general and on smooth curves.
• Picard functor.
• Weil's theorem on ``integrating'' rational group laws, see this beautiful paper by Edixhoven and Romagny.
• The Picard scheme of a curve.
• Abelian varieties.
• Applications and further remarks.

Problem sets. Hand the solutions to me personally, or slide them under my door in the building, or email them to me. Some of the exercises will be impossible, so it should not be your goal to do each and every one of them. Moreover, these exercises are not always doable purely with the material discussed in the course -- sometimes you'll have to look up things online or in books and use what you find.

1. Due Monday 01-27-2025: Do 4 of the exercises from Section Tag 028O.
2. Due Monday 02-3-2025:
   1. Exercise 078S.
   2. Exercise 02AE (you will need to look up pullback of modules and quote results about it).
   3. Exercise 0DT4.
   4. Describe a functor F from the category of rings to the category of sets such that F is not isomorphic the functor of points of a scheme (and please explain why).
   5. Consider the functor F from the category of rings to the category of sets which sends a ring R to the underlying set. Show that this functor is the functor of points of a scheme and tell me what the scheme is.
   6. Same question as before but sending R to the set of units in R.
3. Due Monday 02-10-2025:
   1. In your own words, explain why every scheme has a basis for its topology consisting of affine open subschemes.
   2. Let (X, O_X) be a scheme. Let p : Y → X be a covering space (see this). Show that the pair (Y, p^{-1}O_X) is a scheme.
   3. Give an example of a scheme (X, O_X) such that there exists a nontrivial covering space p : Y → X.
   4. Let k be any field. Show that the topological space X = Spec(k[x]) is connected. Is X path connected too?
   5. Do exercises 029Q and 029R. Hint: reduce it to an algebra question.
   6. Look up the notion of the category of "schemes over a given scheme S" and the notion of the category of "schemes over a ring R" or the category of "schemes over a field k". Explain how we may view the schemes P^1_R and A^1_R constructed in the lectures as schemes over R. Describe the morphisms P^1_R → A^1_R in the category of schemes over R (feel free to assume R is your favorite ring or field, but not the 0 ring of course).
4. Due Wednesday 02-19-2025:
   1. 02FK. Hint: try to divide part (1) into steps: (a) reduce to the case where X is affine, (b) translate the problem in the affine case into commutative algebra, and (c) solve the commutative algebra problem.
   2. 069Y. Hint: you can make an affine example and you can even make an example where k is the real numbers and k' is the complex numbers.
   3. 07DL. Remark: this is a version of Noether normalization for affine plane curves.
   4. 07DM. Hint: the complement of an affine plane curve is affine.
   5. 09TY. Exercise to help you think about finite type ring maps.
5. Due Wednesday 02-26-2025:
   1. Look at the first 3 definitions in Tag 0280 and then do parts (6) and (7) of 02E9. You can use parts (1) -- (5) and (9) in your answer; indeed, some people use part (9) as the definition of the topology on Proj(R).
   2. Let k be a field. Using the definition of P^1_k in our lectures, find a homeomorphism between Proj(k[T_0, T_1]) and our P^1_k.
   3. Do 02EE.
   4. Do 069U.
   5. Try 069X.
6. Due Wednesday 03-05-2025:
   1. Let k be an algebraically closed field. Let P_1, ..., P_r in k[x_1, ..., x_n] be homogeneous of positive degree. If r < n show that there exists a nonzero vector a = (a_1, ..., a_n) in k^n such that P_1(a) = ... = P_r(a) = 0. [Possible ways to solve this: (a) find it in a book and quote it, (b) use the Hilbert Nullstellensatz and a bit of dimension theory, (c) relate it to a problem in projective space and then solve that or find a reference for that, (d) your own idea here.]
   2. Let f(x_1, ..., x_n) be a polynomial with complex coefficients of total degree d. Assume that f(0, ..., 0) = 0. Show that if d is small relative to n, then there is a line in C^n passing through (0, ..., 0) completely contained in the hypersurface f = 0. How small does d have to be? [Hint: use the previous exercise.]
   3. Try your hand at 02FE. It is a special case of the relation between smoothness and formal smoothness translated into algebra.
   4. Give an example of an algebraically closed field k and an irreducible polynomial f in k[x, y] such that X = Spec(k[x, y]/(f)) has exactly 3 singular points, i.e, points where X is not smooth. Alternative: prove abstractly that such a thing exists.
   5. Let k be your favorite field. Let f in k[x, y] of total degree d such that X = Spec(k[x, y]/(f)) has a finite number of singular points (if f is square free and k has characteristic zero, this is automatic). Can you guess an upper bound on the number of singular points? Do you have a heuristic argument for your guess? Can you actually prove some upper bound? Do you have an example of an f where X has a lot of singular points?
7. Due Wednesday 03-12-2025:
   1. Let r, n be positive integers. Prove there exists an integer N with the following property: if we have a field k and a surjection k[x₁, ..., x_r] → B of k-algebras where B has dimension n over k, THEN the images of the monomials x^E = x₁^e₁ ... x₁^e₁ with 0 ≤ e_i ≤ N span B as a k-vector space. (Hint: prove that the image of x_i in B satisfies a small degree monic polynomial.)
   2. 02A8; to do this excercise, if you are not so familiar with projective 2 space, I suggest reading Ravi Vakil's "Rising Sea", Section 4.5.2.
   3. Do as much as you can of 02FJ
   4. Try 0G1&.
8. Due Wednesday 03-26-2025: SAME SET AS 03-12-2025. Hand it in if you haven't yet.
9. Due Wednesday 04-02-2025:
   1. Look up Cech cohomology. Then open up Section Tag 02AO and do the unique problem in this section.
   2. Do 4 or 5 problems from Section Tag 0D8P.
10. Due Wednesday 04-09-2025:
    1. Do Tag 078W.
    2. Do Tag 0DB5.
    3. Do 1 or 2 exercises from Section Tag 0DCD.
11. Due Wednesday 04-16-2025:
    1. Do Tag 02A1. It is enough to do this for n = 1.
    2. Do Tag 0GYD.
    3. Let (E, O) be an elliptic curve over an algebraically closed field. Explain why the map E = \underline{Hilb}^1_{E/k} → \underline{Pic}^1_{E/k} is an isomorphism in this case using the discussion in the lectures. (For example we already discussed why there exists such a morphism. If you can't show it is an isomorphism of schemes, then at least argue it is bijective on points.) Conclude that there is an isomorphism of varieties E → \underline{Pic}^0_{E/k} sending O to O_E. (Thus we see that E has a group law as stated in the lectures.)
    4. Let A be a ring and let a in A be an element. Consider the ring B = A[x, 1/(ax + 1)]. Show there is a (unique) A-algebra homomorphism μ : B → B ⊗_A B with μ(x) = a x⊗x + x ⊗ 1 + 1 ⊗ x. Prove that G = Spec(B) becomes a group scheme over S = Spec(A) with m = Spec(μ) as the group law.
    5. What is the analogue of the circle group in algebraic geometry? In other words, find an algebraic group over the real numbers whose real points give the circle with its group structure.
12. Due Wednesday 04-23-2025: Same as HW 11.

PS: To find the webpages for previous semesters, please visit this page