MA746 Final, due Tuesday, May 8
1. Let Xbe a curve over an algebraically closed field k(meaning a non-
singular projective variety of dimension 1). Assume that the genus of
Xis 2. You may also assume that the characteristic of kis 0.
(a) Let Kbe the canonical divisor of X. Show that `(K) = 2 and
that Kdetermines a morphism f:X→P1of degree 2.
(b) Use the Riemann-Hurwitz formula to show that fis branched over
six points in P1, with ramification degree 2 at each one.
(c) Show that Xdoes not admit a closed immersion into P2.
(d) Show that if Dis a divisor of degree 5 on X, then Dinduces a
closed immersion X→P3.
2. For this exercise, you may use the following fact: Let ri≥2 be a finite
collection of integers, and let R=Pi(1 −1/ri). Then if R > 2, then
R−2≥1/42.
Let Xbe a curve of genus g≥2 over an algebraically closed field of
characteristic 0, and let Gbe a finite group of automorphisms of X.
Assume the existence of a quotient curve f:X→Y, meaning that
the fibers of fare exactly the orbits of G. Use the Riemann-Hurwitz
formula to show that #G≤84(g−1).
(The bound is sharp for infinitely many g. For instance, if Xis the Klein
quartic x3y+y3z+z3x= 0, then Xhas genus 3, and the automorphism
group of Xis isomorphic to PSL2(F7), a simple group of order 168.)
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