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.
- Read a bit about finite and integral ring maps.
For example, take a look at Sections
Tag 0562 and
Tag 00GH.
- Read a bit about Noether Normalization.
For example, take a look at Section
Tag 00OW.
- Read a bit about the spectrum of a ring.
For example, read the proof of
Lemma Tag 00E0
- Read a bit about localization.
For example, take a look at Section
Tag 00CM.
- 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.
- 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.
- Let k be a field. Prove that k[x, y] is not isomorphic to k[x, y, z].
- 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].
- 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.
- Read the parts of Section
Tag 00HU
related to the material we've lectured about (and follow the links).
- Read about local rings in Section
Tag 07BH.
- Read about Nakayama's lemma in Section
Tag 07RC.
- The Chinese remainder theorem is Lemma
Tag 00DT.
- Let k be a field. Let k[[t]] be the power series ring over k.
Show that k[[t]] is a local ring.
- Give an example of (1) a local ring with 2 prime ideals and (2)
a local ring with 3 prime ideals.
- 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).
- 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).
- Let k be a field. Find the minimal prime ideals of
k[x, y, z]/(xy, xz, yz).
- 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.
- Read about Krull dimension in Section
Tag 0054
- Read about irreducible components in Section
Tag 004U
- Extra reading: read about connected components in Section
Tag 004R
and note how similar this is to the section about irreducible components.
- Read the first few results about homomorphisms and dimension in Section
Tag 00OG
- Read a little bit about Artinian rings in Section
Tag 00J4
- Read a bit about Noetherian rings from Section
Tag 00FM
- 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
I^{n} is not zero).
- 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).
- 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].
- Give an example of an Artinian ring which is not an algebra
of finite type over a field.
- 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.
- Read about dimension of Noetherian local rings. For example read Section
Tag 00KD but skip
the material about the function d(-) for now.
- Read about Hilbert Nullstellensatz. For example read Section
Tag 00FS.
- Read about transcendence degree of field extensions in Section
Tag 030D
- Read about integral ring extensions (you probably already have)
for example in Section
Tag 00GH
- 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)?
- 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)?
- 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).
- 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.
- 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?
- 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.
- Read a bit about normal domains in Section
Tag 037B.
- Read about Noetherian graded rings in Section
Tag 00JV.
- 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.
- Given an example of a domain R such that the integral closure of
R in its fraction field is not finite over R.
- 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.
- 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?
- 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?
- 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.
- Read the material on
projective planes,
projective lines,
conics, and
morphisms
explained in the website for my (old) REU project.
- 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.)
- 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).
- 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.
- 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.
- 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.
- Read a bit about valuation rings, for example in Section
Tag 00I8.
- 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
- Give a basis for L(D) when D = 2 v_0 + 3 v_1.
- Give a basis for L(D) when D = 2 v_0 + 2 ∞.
- 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.)
- 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.
- 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.
- 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) ∈ k^{2} | f(s, t) = 0}.
Recall that the maximal ideals of A correspond 1-1 with
points of the curve C.
- 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.)
- Give an example to show this is false when C is singular.
- 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).
- How many valuations of K/k are not centered on A?
- What would you guess is the number
of ``missing'' valuations when you have a general irreducible
f ∈ k[x, y]?
- 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.
- 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.
- 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.
- Conversely, show that every square free f ∈ k[x] gives rise
to a hyperelliptic curve in this way.
- 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).
- 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.
- 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).
- 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.)
- 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.
- 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.
- What is the genus of (the function field of) the curve
D in the first exercise?
- 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.
- Determine a sharp upper bound for the number of intersection
points of a plane in P^3 with the curve D.
- 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
- find an equation for its image D', and
- show that most points in D' have exactly one preimage in D.
- What is the degree of D'?
- How many singular points does D' have?
- What is a guess for the genus of D?
- Twelth problem set due Tuesday November 26.
- Read about sheaves, for example in the
chapter about sheaves
of the Stacks project.
- In particular, read about sheaves on a basis for a topological space in
Section Tag 009H.
- Start reading about schemes, in Hartshorne, in Shafarevich, in
Ravi's FOAG,
or in the
chapter about schemes
of the Stacks project.
- 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.
- 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.
- 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.
- Prove that the spectrum of a ring is a quasi-compact topological space.
Deduce that every standard open is quasi-compact.
- Look at and do some of the exercises in Hartshorne about
sheaves on spaces.
- Thirteenth problem set due Tuesday December 3.
- Look up what it means for a morphism of schemes to be surjective.
- Give an example of a surjective morphism of schemes X ---> Y
which does not have a right inverse.
- Read about morphisms into affine schemes, for example
Tag 01I1.
Please just focus on what the statement says and not the proof.
- Read about the Yoneda lemma in the setting of schemes, see
Tag 001L and
the introduction to
Tag 001JF.
- 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.
- Enjoy your holiday!
- Fourteenth problem set due first lecture next semester.
- Read in the book Algebraic Geometry by Hartshorne, Chapter II,
Sections 1, 2, 3, 4.