Math 419/819  Homework Assignment 2  September 30, 2003

due date: October 7, 2003

The answers to the following problems should again be short – no long essay is required!

1. Several times in class I’ve mentioned that the only functions ℙ^{1} ⟶ ℂ which are holomorphic in all charts (a global holomorphic function) are the constant functions. The purpose of this question is to see why this is true.

Suppose that f : ℙ^{1} ⟶ ℂ is a global holomorphic function. In terms of the chart V_{1}, that means we should get a holomorphic function f_{1} : V_{1} ⟶ ℂ, which will therefore have a power series expansion:

f_{1}(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + a_{4}z^{4} + · · · · · ·

the expansion has infinite radius of convergence since the function is defined on all of V_{1} ≅ ℂ.

Similarly, in terms of the chart V_{2}, we get a holomorphic function f_{2} : V_{2} ⟶ ℂ, with power series expansion

f_{2}(w) = b_{0} + b_{1}w + b_{2}w^{2} + b_{3}w^{3} + b_{4}w^{4} + · · · · · ·

On the overlap, the charts are related by the transition function w = 1/z (or, z = 1/w). Since the functions f_{1}(z) and f_{2}(w) must match up on the intersection of the two charts, explain why this forces both f_{1} and f_{2} (and hence f) to be constant.

The purpose of the next two questions is to think about the basic details of ramified covers, and to practice using the Riemann-Hurwitz formula.

2. Suppose that π : X_{1} ⟶ X_{2} is a map of degree d between Riemann surfaces.

(a) For any point p of X_{1}, explain why the ramification index k_{p} cannot be bigger than d, i.e., that k_{p} ≤ d.

(b) If d = 2 (a double cover) explain why a point p of X_{1} is either not a ramification point, or has ramification index exactly 2.

(c) Again in the case that d = 2 explain why the number of branch points (on X_{2}) is the same as the number of ramification points (on X_{1}).

Parts (b) and (c) show that for degree 2 covers, the topological data (the number and type of ramification points) of a map π : X_{1} ⟶ X_{2} is given just by knowing the number of ramification or branch points. Use this to answer the following two questions.

(d) Suppose that π : X_{1} ⟶ X_{2} is a degree 2 map. Show that the number of ramification points is even.

(e) Suppose that π : X_{1} ⟶ ℙ^{1} is a double cover, with 2t branch points. What is the genus of X_{1}?.

3. Use the Riemann-Hurwitz formula to find the genus of X_{1}, the genus of X_{2}, or the number of ramification points, as required.

(a) π : X_{1} ⟶ ℙ^{1} is a degree 3 cover, with two ramification points, both with ramification index k_{p} = 3. Find the genus of X_{1}.

(b) π : X_{1} ⟶ ℙ^{1} is a degree 3 cover, with three ramification points, all with ramification index k_{p} = 3. Find the genus of X_{1}.

(c) π : X_{1} ⟶ X_{2} is a map of degree d, X_{1} has genus 1, and there are no ramification points. Find the genus of X_{2}.

(d) X_{1} is of genus g, X_{2} is of genus 1, the map π : X_{1} ⟶ X_{2} is of degree d, and all ramification points p in X_{1} are of index 2. Find the number of ramification points (the answer turns out, in this case, not to depend on the degree d).

Can you think of a map X_{1} ⟶ ℙ^{1} satisfying the description in part (a)?