Today as part of my qualifying exam studying I want to sketch an example which occurs in chapter 4 of the tome of algebraic geometry written by everyone’s favorite shakuhachi player.
We are interested in studying smooth proper curves over an algebraically closed field and it turns out that an interesting invariant (especially for curves of higher genus) is whether or not a curve admits a nontrivial degree two morphism to . We call such curves hyperelliptic if their genus is at least .
One the one hand, if a curve is hyperelliptic we get an interesting map to work with which among other things immediately tells us that the functional field is a degree two extension of . On the other, if is not hyperelliptic, then the canonical divisor of is very ample and so gives a canonical embedding .
Lets see how this second property comes about. Recall that a divisor on is very ample precisely when for any two points . In particular, removing a point from a divisor drops the dimension of the local system by at most and very ampleness is equivalent to the dimension drop being the maximum possible for every pair of points. Picking any two and applying Riemann-Roch gives
Since , we see that is zero for all exactly when is very ample and that if this is not the case, we have for some pair of points and this gives a degree two morphism to .
Now, since the canonical divisor of an elliptic curve is degree it’s clearly impossible for it to be very ample so all elliptic curves admit a morphism to . In fact, we knew this anyway because Weierstrass form gives a very explicit degree extension of .
For genus and so very ampleness would imply that which is absurd. Thus all genus two curves are hyperelliptic.
We’d like to find a non-hyperelliptic curve and so the next place to look is genus three. Here, the situation is not immediately impossible, and in fact we will find examples. The image of a canonical embedding would have to be a quartic curve in and so we consider such curves. In fact, we will see that if is a nonsingular quartic, then and so all such curves are non-hyperelliptic.
Since is nonsingular, we have an exact sequence of sheaves on :
Then here is a nice observation. If is an exact sequence of locally sheaves of rank respectively, then splits up into a filtration so that . Thus, if we take the top exterior power our filtration has only one nonzero term, and this gives .
Tensoring both sides with gives
But the ideal sheaf and so and so our final formula is
The right-hand side is and so we get what we wanted.