## Hurwitz’s Theorem

Ok, back to curves. We’d wandered a bit in the direction of this topic before, having discussed Bezout’s Theorem and the Riemann-Roch Theorem. Today we’ll talk about the Hurwitz formula, also called the Riemann-Hurwitz formula. It’s a rather nice result, and combined with Riemann-Roch can be used to prove a huge amount about curves and maps between curves.

We’re going to be very specifically working over $\mathbb{C}$ here, and will use some analytic tools. The first one of which is that, given a map $F:X\to Y$ of smooth curves which is nonconstant, and pick a point $p\in X$. We can choose a coordinate $w$ on $X$ and $z$ on $Y$ in a neighborhood of $p$ and of $F(p)$ (note, neighborhood in the usual topology, not the Zariski topology) such that $z=w^n$ for some $n$. Even better, the number $n$ is uniquely determined by the map. This is called the local normal form for a map between curves. We define the multiplicity of a map $mult_p F$ to be this $n$, and we say that $p$ is a ramification point for $F$ if $mult_p F\geq 2$. The images of ramification points are called branch points.

Now, the ramification and branch points must form a discrete set, and so if we have a projective curve (ie, a compact Riemann surface) there are only finitely many.

Now let’s take a moment and consider what the geometry of a ramification point is. Say just simple ramification, a point where $mult_p F=2$. Well, the map locally looks like $w^2=z$ projected onto $z$. So near the ramification point, it’s just two sheets coming together, but there it needs to be counted twice, just like the intersection of the parabola $y=x^2$ and the line $y=0$ needs to be counted twice in Bezout’s Theorem. A related way to look at it is given by the fact that the number $d=\sum_{p\in F^{-1}(y)} mult_p(F)$ doesn’t depend on $y\in Y$. So then most points will have $d$ preimages, but the branch points have fewer because some points are counting extra. We call this $d$ the degree of $F$.

To show that these definitions can be useful, we note that if a map is of degree 1, then it’s a one-to-one map, unramified. Thus, $F$ is an isomorphism between $X$ and $F(X)$. However, if we restrict to the projective case for both $X$ and $Y$, it’s better, because the image must be closed, nonconstant, and connected. Thus, $F(X)=Y$, and so $F$ is an isomorphism $X\cong Y$. So that means that maps of degree 1 are isomorphisms, because we’re going to restrict to projective curves (well, complete curves, which are still compact Riemann surfaces, but all happen to be projective because curves are nice and as a consequence of Riemann-Roch) and so we can get a really nice corollary immediately. Say a curve has a rational function on it with just a single simple pole. Then automatically the curve is $\mathbb{P}^1$. The rational function extends to a morphism $X\to \mathbb{P}^1$, and it sends a single point to $\infty$ with multiplicity 1, so the degree is one. Now, we’ll state Hurwitz’s Theorem:

Hurwitz’s Theorem: Let $F:X\to Y$ be a nonconstant morphism of smooth complete curves. Then $2g(X)-2=\deg(F)(2g(Y)-2)+\sum_{p\in X} (mult_p F-1)$.

The last term is a finite sum, and is called the ramification number of the map. It’s also the degree of a divisor $R=\sum_{p\in X} (mult_p F-1)p$, called the ramification divisor. Before proving the theorem, let’s talk a bit more about ramification divisors and other ways to obtain the same divisor.

Way back, we talked about the relative cotangent sheaf, $\Omega_{X/Y}$ which depends on our morphism. In fact, because $R$ is a closed subscheme of $X$, we can look at its structure sheaf, and even push it forward onto $X$. Call this $\mathscr{O}_R$. It turns out to be isomorphic to $\Omega_{X/Y}$, which means that we can use $\Omega_{X/Y}$ to compute ramification numbers of points, though right now we want this isomorphism only because there’s a short exact sequence $0\to f^*\Omega_Y\to \Omega_X\to \Omega_{X/Y}\to 0$. We can tensor this with $\Omega_X^{-1}$, and we get an exact sequence $0\to f^*\Omega_Y\otimes \Omega_X\to \mathscr{O}_X\to \mathscr{O}_R\to 0$. So then $f^*\Omega_Y\otimes \Omega_X^{-1}$ is isomorphic to the kernel of the map $\mathscr{O}_X\to \mathscr{O}_R$, which is just the ideal sheaf of $R$.

Now, for any divisor $D$, the ideal sheaf defining it is actually going to be $\mathscr{O}(-D)$. So here we have $f^*\Omega_Y\otimes\Omega_X^{-1}=\mathscr{O}(-R)$. Tensoring with $\mathscr{O}(R)\otimes\Omega_X$, we get $f^*\Omega_Y\otimes \mathscr{O}(R)=\Omega_X$, which in terms of divisors tells us that $K_X=f^*K_Y+R$.

Now, Hurwitz is a trivial consequence of this fact, because we take degrees. We know that $\deg K_X=2g(X)-2$, and that $\deg f^*D=\deg(f)\deg (D)$, so this becomes $2g(X)-2=\deg(F)(2g(Y)-2)+\deg R$, as desired.

Now we should explore some consequences of this fact. This does show immediately that if we have a nonconstant morphism $f:X\to Y$ of curves, then $g(X)\geq g(Y)$, so there aren’t any interesting maps $\mathbb{P}^1\to X$ unless $X$ is $\mathbb{P}^1$ itself.

Hurwitz gives us a new way to compute the genus of a plane curve. Let $C\subset\mathbb{P}^2$ be a plane curve, and look at the morphism given by projection onto a line. This is a map $C\to \mathbb{P}^1$, and so Hurwitz becomes $2g-2=d(-2)+\deg R$. Assuming that we choose the point of the projection to be nice enough (ie, it doesn’t lie on any line that is tangent to the curve at two points, etc) then $\deg R=d(d-1)$, so we get $2g-1=d(d-1)-2d$. The $d(d-1)$ is just the number of simple tangent lines of $C$ which pass through the point we are projecting from.

Another thing that Hurwitz tells us is that for $g(X)=g(Y)\geq 2$, we always have isomorphisms, never anything else. If we drop the ramification divisor, we get an inequality $2g(X)-2\geq d(g(Y)-2)$, which is really $(2g-2)\geq d(2g-2)$. The only way this can happen with $d$ a positive integer is if $d=1$, so we have an isomorphism to begin with.

Next time we’ll use Hurwitz’s Theorem to prove Hurwitz’s Theorem on Automorphisms, which bounds the size of a finite group that can act on curves of genus $\geq 2$.