Algebraic Geometry

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

The plan of this semester course in algebraic geometry is twofold. We start doing some enough commutative algebra to get a fairly complete overview of dimension theory. Then we'll briefly discuss some further topics in commutative algebra. At this point we'll switch track and start talking about (old school) algebraic varieties and in particular curves. Hopefully we'll be able to mention some of the more interesting aspects of curve theory, such as linear systems.

Lecture notes taken by Nilay Kumar and Matei Ionita. You can find the source for these notes in one of his github repositories.

There is not going to be a book associated to the course. All the commutative algebra will be 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 September 10.
  1. Read a bit about finite and integral ring maps. For example, take a look at Sections Tag 0562 and Tag 00GH.
  2. Read a bit about Noether Normalization. For example, take a look at Section Tag 00OW.
  3. Read a bit about the spectrum of a ring. For example, read the proof of Lemma Tag 00E0
  4. Read a bit about localization. For example, take a look at Section Tag 00CM.
  5. Let A = k[x, y]/(x^2y^4 - x^4y^2 + 1) where k is a field. Construct a k-algebra map as in Noether Normalization for A. Very briefly explain why it works.
  6. Describe all prime ideals of C[x, y]/(xy) where C is the field of complex numbers. Just list them in some way and explain briefly why they are primes and why you've got all of them.
  7. Let k be a field. Prove that k[x, y] is not isomorphic to k[x, y, z].
  8. Let k be a field. Suppose A is a k-algebra and f is a nonzerodivisor of A such that k[x, y] is isomorphic to A_f as k-algebra. Show that A is isomorphic to k[x, y].
  9. Let A be a ring and let f be an element of A. Show that A_f is as an A-algebra isomorphic to A[x]/(fx - 1).
• Second problem set due September 17. Please make your answers concise.
  1. Read the parts of Section Tag 00HU related to the material we've lectured about (and follow the links).
  2. Read about local rings in Section Tag 07BH.
  3. Read about Nakayama's lemma in Section Tag 07RC.
  4. The Chinese remainder theorem is Lemma Tag 00DT.
  5. Let k be a field. Let k[[t]] be the power series ring over k. Show that k[[t]] is a local ring.
  6. Give an example of (1) a local ring with 2 prime ideals and (2) a local ring with 3 prime ideals.
  7. Let R = k[[t]] where k is a field. Give an example of a module M over R such that M = tM (in other words, a module which contradicts the conclusion of Nakayama's lemma).
  8. Let R = C[x] be the polynomial ring over the complex numbers. Let m_n, n = 1, 2, 3, ... be an infinite sequence of pairwise distinct maximal ideals of R. Show that R does not surject onto the product of the rings R/m_n (contradicting the conclusion of the Chinese remainder theorem).
  9. Let k be a field. Find the minimal prime ideals of k[x, y, z]/(xy, xz, yz).
  10. More difficult: Let R be a ring and let p and q be prime ideals of R. Show that either one can find disjoint standard opens D(f) and D(g) with p ∈ D(f) and q ∈ D(g), or one can find a prime ideal r contained in both p and q. (At the end of the 4th lecture you will have enough tools to solve this question in a fairly easy manner, but finding the argument may be a bit of a puzzle.)
• Third problem set due September 24. Try to be succint but clear.
  1. Read about Krull dimension in Section Tag 0054
  2. Read about irreducible components in Section Tag 004U
  3. Extra reading: read about connected components in Section Tag 004R and note how similar this is to the section about irreducible components.
  4. Read the first few results about homomorphisms and dimension in Section Tag 00OG
  5. Read a little bit about Artinian rings in Section Tag 00J4
  6. Read a bit about Noetherian rings from Section Tag 00FM
  7. Find a ring A and an ideal I such that I is generated by countably many elements f_1, f_2, f_3, ... such that f_i^2 = 0 but such that I is not a nilpotent ideal (in other words for all n > 0 the ideal Iⁿ is not zero).
  8. Let A ⊂ B be an extension of domains. Let K be the fraction field of A and L be the fraction field of B, so that we have an extension of fields K ⊂ L. Show that if (a) B is a finite type A-algebra and (b) L is a finite extension of K, then the image of Spec(B) ---> Spec(A) contains a nonempty open subset of Spec(A).
  9. Let k be a field. Let f, g ∈ k[t] be two polynomials in avariable t with coefficients in k. Show that there exists a nonzero two variable polynomial P ∈ k[x, y] such that P(f, g) = 0 in k[t].
  10. Give an example of an Artinian ring which is not an algebra of finite type over a field.
  11. More difficult: Give an example of a ring A, a prime ideal p of A, and an integer n such that the nth symbolic power of p is not equal to the nth power of p.
• Fourth problem set due October 1.
  1. Read about dimension of Noetherian local rings. For example read Section Tag 00KD but skip the material about the function d(-) for now.
  2. Read about Hilbert Nullstellensatz. For example read Section Tag 00FS.
  3. Read about transcendence degree of field extensions in Section Tag 030D
  4. Read about integral ring extensions (you probably already have) for example in Section Tag 00GH
  5. What is the dimension of the local ring of k[x, y, z]/(x^2y^2z^2, x^3y^2z) at the maximal ideal (x, y, z)?
  6. What is the dimension of the local ring of k[x, y, z]/(x^3 - y^2, x^5 - z^2, y^5 - z^3) at the maximal ideal (x, y, z)?
  7. Let k be a field. Let f ∈ k[x, y] be a polynomial. Let a, b ∈ k be elements such that f(a, b) = 0. Let m = (x - a, y - b) be the corresponding maximal ideal in the ring A = k[x, y]/(f). Prove that A_m is a regular local ring if and only if one of df/dx, df/dy doesn't vanish at (a, b).
  8. Definition. Let k be an algebraically closed field. Let f ∈ k[x, y] be a polynomial. We say that C = {(a, b) ∈ k^2 | f(a, b) = 0} is the curve associated to f. We say P = (a, b) is a nonsingular point if the equivalent conditions of the previous exercise hold.
  9. Let k = C be the field of complex numbers. What are the singular points of the curve C defined by f = x^n + y^n + 1, f = xy^2 + x^2y, f = x^2 - 2x + y^3 - 3y^2 +3y?
  10. Let k be an algebraically closed field. Let f ∈ k[x, y] be a squarefree polynomial of degree ≤ d. What is the maximum number of singular points the associated curve C can have? Start with d = 1, 2, 3,... and make a guess for the general answer. To prove it in general is too hard right now.
• Fifth problem set due October 8.
  1. Read a bit about normal domains in Section Tag 037B.
  2. Read about Noetherian graded rings in Section Tag 00JV.
  3. Let k = C be the field of complex numbers. Compute the integral closure of the domain k[x, y, z]/(z^6 - x^2 y^3) in its fraction field.
  4. Given an example of a domain R such that the integral closure of R in its fraction field is not finite over R.
  5. Let k be a field. Let M be a graded k-vector space. Then the Hilbert function is the function which assigns to an integer n the dimension of the nth graded part of M. Then Hilbert polynomial, if it exists, is the polynomial whose value at n is equal to the Hilbert function for all n >> 0.
  6. Let k be a field and B = k[x, y] with grading determined by deg(x) = 2 and deg(y) = 3. Compute the Hilbert function of B. Is there a Hilbert polynomial in this case?
  7. Let k be a field and B = k[x, y]/(x^2, xy) with grading determined by deg(x) = 2 and deg(y) = 3. Compute the Hilbert function of B. Is there a Hilbert polynomial in this case?
  8. Let k be a field and B = k[x, y, z]/(x^d + y^d + z^d) with grading determined by deg(x) = deg(y) = deg(z) = 1. Compute the Hilbert function of B. Is there a Hilbert polynomial in this case?
• Sixth problem set due October 15.
  1. Read the material on projective planes, projective lines, conics, and morphisms explained in the website for my (old) REU project.
  2. Do Exercise 1 from the page about projective planes on the REU website: Prove that an axiomatic projective plane has the same number of points as lines. (You get extra points for noticing the missing axiom and fixing.)
  3. Do Exercise 8 from the page about projective lines on the REU website: Show that if P, Q, R are three pairwise distinct points on P^1 then there exists a matrix A which determines a map P^1 ---> P^1 mapping P, Q, R to (1 : 0), (0 : 1), and (1 : 1).
  4. Do Exercise 10 from the page about conics on the REU website: Find a field K and a conic as defined above without any points.
  5. Do Exercise 17.a from the page about morphisms on the REU website: Prove that a degree two morphism P^1 ---> P^2 maps onto either a line or a conic.
  6. Let k be an algebraically closed field. Let k(t) be the field of rational functions over k. Let k(t) ⊂ K be a finite extension. Prove or look up the proof of the following statements: (a) the integral closure of k[t] in K is finite over k[t], (b) for every discrete valuation v on k(t) there are finitely many discrete valuations w_i on K whose restriction to k(t) is e_iv for some integer e_i, and (c) we have ∑ e_i = [K : k(t)].
• Seventh problem set due October 22.
  1. Read a bit about valuation rings, for example in Section Tag 00I8.
  2. Let k be an algebraically closed field. Let K = k(t). Denote v_c = ord_{t = c} the valuation corresponding to c in k. Denote ∞ the valution corresponding to the point at infinity. With this notation
     1. Give a basis for L(D) when D = 2 v_0 + 3 v_1.
     2. Give a basis for L(D) when D = 2 v_0 + 2 ∞.
  3. Assume k does not have characteristic 2. Let K be the degree 2 extension of k(t) defined by y^2 = f(t) for some cubic squarefree polynomial f. Find all the discrete valuations on K/k. In other words, analyze the structure of these discrete valuations as we did in the class for the field k(t), but try to use as much as you can the lemmas from the lectures. (If you like you can pick a specific f and a specific k.)
  4. With k and K as in the previous question, show that K is not a purely transcendental extension of k. (Hint: show that there exists an effective degree 1 divisor D with dim L(D) = 1 and observe that this doesn't happen for the field k(t).)
• Eigth problem set due October 29.
  1. Definition: Let A be a domain with fraction field K. We say a discrete valuation v is centered on A if v(a) ≥ 0 for all a ∈ A.
  2. Let k be an algebraically closed field. Let A = k[x, y]/(f) where f is an irreducible polynomial. Let K be the fraction field of A. Let C = {(s, t) ∈ k² | f(s, t) = 0}. Recall that the maximal ideals of A correspond 1-1 with points of the curve C.
     1. Show that if every point of C is nonsingular, then the valuations of K centered on A are in 1-1 correspondence with points of C. (Hint: Above you showed that the local rings of A are regular at nonsingular points. You may use that a regular local ring of dimension 1 is a discrete valuation ring and hence gives rise to a discrete valuation, see for example Lemma Tag 00PD.)
     2. Give an example to show this is false when C is singular.
  3. Let k = C be the field of complex numbers. Let f = 1 + x^n + y^n for some positive integer n. Let K be the fraction field of A = k[x, y]/(f).
     1. How many valuations of K/k are not centered on A?
     2. What would you guess is the number of ``missing'' valuations when you have a general irreducible f ∈ k[x, y]?
     3. Give an example to show that your guess is wrong!
• Ninth problem set due Thursday November 7. Hyperelliptic curves. Throughout this problem set we assume that the base field k has characteristic not equal to 2.
  1. Definition. A hyperelliptic curve is a curve C whose function field K is a degree 2 extension of a purely transcendental extension of the base field k.
  2. Show that every hyperelliptic curve is birational to a curve of the form y^2 = f(x) where f ∈ k[x] is a monic square free polynomial.
  3. Conversely, show that every square free f ∈ k[x] gives rise to a hyperelliptic curve in this way.
  4. Give an example to show that two distinct monic square free f ∈ k[x] can lead to isomorphic curves (for us this means that the function fields are isomorphic as extensions of k).
  5. Given a hyperelliptic curve C : y^2 = f(x) as above let D be the zero divisor of x on C. The degree of D is 2. Show that l(D) = 2 if g > 0.
  6. Show that a curve C which has a divisor D with deg(D) = 2 and l(D) = 2 is hyperelliptic (we may discuss this in class).
  7. Let C : y^2 = f(x) as above. Consider the differential form ω = dx. Compute its zeros and poles on C and as a consequence compute the genus of C. (The cases deg(f) even or odd are slightly different. Just do one of the two cases.)
  8. Extra credit. Let C be a curve of genus at least 2. Show that if D and D' are divisors with deg(D) = deg(D') = 2 and l(D) = l(D') = 2, then D is rationally equivalent to D'.
• Tenth problem set due Tuesday November 12. Catch up with reading and previous problem sets.
• Eleventh problem set due Tuesday November 19.
  1. Let C be the field of complex numbers. Find an irreducible polynomial F in C[X_0, X_1, X_2] homogeneous of degree 4 such that the curve D = V(F) has 3 distinct singular points.
  2. What is the genus of (the function field of) the curve D in the first exercise?
  3. Working over C consider the curve D in P^3 defined by the equations X_0^2 + X_1^2 + X_2^2 + X_3^2 = 0 and X_0^3 + X_1^3 + X_2^3 + X_3^3 = 0.
     1. Determine a sharp upper bound for the number of intersection points of a plane in P^3 with the curve D.
     2. Find an irreducible curve D' in P^2 which is birational to D by projection. Namely, consider the map which sends (X_0, X_1, X_2, X_3) to (X_0, X_1, X_2) which is well defined on D and
        1. find an equation for its image D', and
        2. show that most points in D' have exactly one preimage in D.
     3. What is the degree of D'?
     4. How many singular points does D' have?
     5. What is a guess for the genus of D?
• Twelth problem set due Tuesday November 26.
  1. Read about sheaves, for example in the chapter about sheaves of the Stacks project.
  2. In particular, read about sheaves on a basis for a topological space in Section Tag 009H.
  3. Start reading about schemes, in Hartshorne, in Shafarevich, in Ravi's FOAG, or in the chapter about schemes of the Stacks project.
  4. Without looking at the proof in Hartshorne or elsewhere, show that given a ring A and elements f_i of A which generate the unit ideal, there is an exact sequence 0 --> A ---> prod A_{f_i} ---> prod A_{f_if_j}. In other words, prove the sheaf condition for the structure sheaf of an affine scheme on the basis of standard opens. Of course, only do this exercise if you haven't seen this yet.
  5. Give an example of a ringed space which is not a locally ringed space. You do not have to explain why your example is an example, just present the example clearly.
  6. Give an example of locally ringed spaces X and Y and a morphsm of ringed spaces X ---> Y which is not a morphism of locally ringed spaces. You do not have to explain why you example is an example, just present the example clearly.
  7. Prove that the spectrum of a ring is a quasi-compact topological space. Deduce that every standard open is quasi-compact.
  8. Look at and do some of the exercises in Hartshorne about sheaves on spaces.
• Thirteenth problem set due Tuesday December 3.
1. Look up what it means for a morphism of schemes to be surjective.
2. Give an example of a surjective morphism of schemes X ---> Y which does not have a right inverse.
3. Read about morphisms into affine schemes, for example Tag 01I1. Please just focus on what the statement says and not the proof.
4. Read about the Yoneda lemma in the setting of schemes, see Tag 001L and the introduction to Tag 001JF.
5. Let X_1, X_2, X_3, ... be a sequence of affine schemes. Show that the functor which sends a scheme T to the infinite product Mor(T, X_1) x Mor(T, X_2) x Mor(T, X_3) x ... is representable by an affine scheme.
6. Enjoy your holiday!
• Fourteenth problem set due first lecture next semester.
1. Read in the book Algebraic Geometry by Hartshorne, Chapter II, Sections 1, 2, 3, 4.