MMA320 Introduction to Algebraic Geometry Autumn 24

This page contains the program of the course: lectures, exercise sessions, homework . Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

2024-12-11: Many of the considerations for the last two lectures is based on the material in chapter 8 of https://dept.math.lsa.umich.edu/~wfulton/CurveBook.pdf . I hope to be able to finish writing my own version of this, but it takes more time than I expected. 

2024-12-10: Here are some problems we will discuss on Friday:

1a. Let C be a curve of genus one. Show that if D is a degree one divisor, then D is linearly equivalent to a unique point on P. 

b. Show that C has a natural group structure, by mapping it into the degree 0 divisors in the Picard group. 

c. Show that the group law on a cubic is associative using this construction (choosing for simplicity the O as an inflection point)

2a. For a map of smooth projective curves f: C -> C' (in characteristic 0) of degree d, prove the Riemann-Hurwitz formula: 

deg K_C = d deg K_(C') +sum_(Q in C) (e_Q-1).

Here e_Q is defined in terms of ramification indices: If t_P is a parameter at P = f(Q), then t_P generates an ideal in O_(C, Q), which is of the form (t_Q)^(e_Q).  

b. Use Riemann--Hurwitz to show that a smooth plane curve of degree d, has g = (d-1)(d-2)/2. In particular any Fermat curve x^d + y^d = 1 not rationally parametrized, for d >= 3.

3. Fixing a basis of L(D)  determines a so called linear system P(l(D)) = |D|. Show that using those functions, one can define a function C -> |D|.  

4. Suppose that the map into the linear system associated to |K_C| is a closed embedding. Study this condition for g =1,2,3,4.  

2024-12-04: I promised I would describe the line between (0:1:0) and a given point (p:q:1). Such a line is a line ax+by+cz=0, with b=0 (the first point lies on the line) and a p +c=0, i.e. a = -c/p. In other words the line is given by -cx/p + cz=0, or simplifying: x = p z. In affine coordinates it is given by x = p, hence a straight line pointing upwards from (p:q:1) as I drew in the class.  

2024-12-03: On Friday, I plan to go through the following exercises: 

0) Study the singular points on x^2 = xy = 0 (i.e. on the affine alg. set corresponding to the ideal (x^2, xy)), which is the line x = 0 with an embedded component at the origin. 

1) Consider the finite field F_p in F_p-bar. These points are characterized by the equation x^p = x. Now, consider a smooth cubic curve with coefficients in F_p, denoted by C. Show that: 

a) The map on points Fr: (x:y:z) -> (x^p: y^p,: z^p) defines a regular map C -> C. 

b) There is a point such that Fr(P) = P. In other words, if you are given a cubic polynomial over Z, with "good reduction modulo p", there is always a solution modulo p. 

2) Consider the linear system of degree d plane curves in P^2. Show that there is an algebraic set in this linear system such that those points correspond to singular degree d plane curves. 

3) Compute this algebraic set for the restriction to the linear system restricted to y^2 = x^3 + ax+ b.

 4) Show that P^1 is not an algebraic group, i.e. there are no algebraic maps P^1 x P^1 -> P^1 such that it behaves like the "addition map" in a group.

5) Show that a smooth cubic curve is not birational to P^1. This is our first example of a curve with k(E) != k(t). 

6) Suppose that F, G, H are homogenous polynomials in 3 variables, F and G not sharing any factors with H. Sloppily write the associated curves by the same letters. Then the intersection numbers satisfy (F G, H) = (F, H) + (G,H).

2024-11-26: On Thursday, I plan to go through the following exercises: 

(might add more later)

2.7. 

- Describe the local rings O_{P^1, a} for all points a in P^1. 

• Go through the main theorem of elimination theory, as below: 
• First reduce to projective and affine spaces. 
• Reformulate the problem by using our version of the projective Nullstellensatz. 
• Then provide the proof that the sets W_s are closed, in two steps: 

a) Show the set of monomials of degree k, in n+1 variables, contains (n + k choose k) elements. 

b) Show that the set W_s is related to the surjectivity of the map S_(d-d1) x ... S_(d-dk) -> S_s, where S_k subset of k[Y1, ... Ym][X0,...Xn], of polynomials which are homogenous of degree k. 

2.23.

2.28. 

2024-11-21: I plan to go through the following exercises tomorrow (some notions will be handled in today's class): 

1) Show that x^n + y^n = 1 is a rational curve for n = 2, but not a rational curve for n >= 3 (we suppose that the characteristic of the field is not positive).

2) Study the local ring of the curve y^2 = x^3+x^2 (the nodal cubic). Show that the local ring of the origin is not a discrete valuation ring. 

3) Show that the previous curve is birational to the affine line. 

4) Classify the discrete valuations on k(t). 

5a) Compute the intersection multiplicity of y^2 = x^3 and y =0 or x = 0.

b) For a plane curve C given by F(x,y) = 0 and P a point on P, what is the smallest intersection multiplicity I(P, C cap L), where L is a line that also passes through P? 

c) This number gives a definition of multiplicity of a point with a curve. Show that it is 1 if and only if the point is non-singular, i.e. if not both partial derivatives of F vanish at this point.

2024-11-13: I plan to go through the exercises 1.19, 1.21, 1.23, 1.32, plus a study of the coordinate rings for Y^2 = X^3 and Y^2= X^2(X+1) on Friday. 

2024-11-11: I have done two of the exercises I didn't have time to do on the exercise session. Please find it here.

Program

See below for last years plan. This year will follow a similar setup.  The lectures will be based on the notes here.

A schedule has been proposed and can be found in TimeEdit. Modifications might occur.

I have tried to make it so the maximum number of people can participate according to the feedback I got. If the situation will change during the coming month, please let me know.

Lectures

The schedule is preliminary.

Week	Chapter	Content
45	1	affine algebraic varieties, resultant
46	1	Nullstellensatz, irreducible components, regular functions
47	2, 3	projective varieties
48	3,4	plane curves, Bezout's theorem
49	4	the group law on a cubic curve, dimension, smoothness.
50	5	Curves and Riemann-Roch

 

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Recommended exercises

Week	Exercises
44	1.1, 1.5, 1.6, 1.8, 1.11, 1.13, 1.14, 1.18, 1.22
45	1.19, 1.20, 1.21, 1.23, 1.25, 1.28, 1.29, 1.30, 1.33
46	2.1, 2.2, 2.3, 2.6, 2.11, 2.14, 2.19, 2.20, 2.21, 2.27, 2.28
47	2.27, 2.28, 3.6, 3.12, 3.13, 3.16, 3.19, 3.20, 3.22, 3.27
48	3.27, 4.1, 4.2, 4.4, 4.5, 4.6, 4.10, 4.12
49	5.2, 5.3, 5.4, 5.6, 5.9
50	6.1, 6.2, 6.4, 6.5, 6.7, 6.8

 

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Homework

You can hand in in any format. You can e.g. scan handwritten solutions.

 

 

 

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