I’ve been trying to learn a little about algebraic curves lately, and genus two is a nice starting point where the general features don’t get too unmanageable, but plenty of interesting phenomena still arise.

0. Introduction

Every genus two curve ${C}$ is hyperelliptic in a natural manner. As with any curve, the canonical line bundle ${K_C}$ is generated by global sections. Since there are two linearly independent holomorphic differentials on ${C}$, one gets a map

$\displaystyle \phi: C \rightarrow \mathbb{P}^1.$

Since ${K_C}$ has degree two, the map ${\phi}$ is a two-fold cover: that is, ${C}$ is a hyperelliptic curve. In particular, as with any two-fold cover, there is a canonical involution ${\iota}$ of the cover ${\phi: C \rightarrow \mathbb{P}^1}$, the hyperelliptic involution. That is, every genus two curve has a nontrivial automorphism group. This is in contrast to the situation for higher genus: the general genus ${g \geq 3}$ curve has no automorphisms.

A count using Riemann-Hurwitz shows that the canonical map ${\phi: C \rightarrow \mathbb{P}^1}$ must be branched at precisely six points, which we can assume are ${x_1, \dots, x_6 \in \mathbb{C}}$. There is no further monodromy data to give for the cover ${C \rightarrow \mathbb{P}^1}$, since it is a two-fold cover; it follows that ${C}$ is exhibited as the Riemann surface associated to the equation

$\displaystyle y^2 = \prod_{i=1}^6 (x - x_i).$

More precisely, the curve ${C}$ is cut out in weighted projective space ${\mathbb{P}(3, 1, 1)}$ by the homogenized form of the above equation,

$\displaystyle Y^2 = \prod_{i = 1}^6 ( X - x_i Z).$

1. Moduli of genus two curves

It follows that genus two curves can be classified, or at least parametrized. That is, an isomorphism class of a genus two curve is precisely given by six distinct (unordered) points on ${\mathbb{P}^1}$, modulo automorphisms of ${\mathbb{P}^1}$. In other words, one takes an open subset ${U \subset (\mathbb{P}^1)^6/\Sigma_6 \simeq \mathbb{P}^6}$, and quotients by the action of ${PGL_2(\mathbb{C})}$. In fact, this is a description of the coarse moduli space of genus two curves: that is, it is a variety ${M_2}$ whose complex points parametrize precisely genus two curves, and which is “topologized” such that any family of genus two curves over a base ${B}$ gives a map ${B \rightarrow M_2}$. Moreover, ${M_2}$ is initial with respect to this property.

It can sometimes simplify things to assume that three of the branch points in ${\mathbb{P}^1}$ are given by ${\left\{0, 1, \infty\right\}}$, which rigidifies most of the action of ${PGL_2(\mathbb{C})}$; then one simply has to choose three (unordered) distinct points on ${\mathbb{P}^1 \setminus \left\{0, 1, \infty\right\}}$ modulo action of the group ${S_3 \subset PGL_2(\mathbb{C})}$ consisting of automorphisms of ${\mathbb{P}^1}$ that preserve ${\left\{0, 1, \infty\right\}}$. In other words,

$\displaystyle M_2 = \left( \mathrm{Sym}^3 \mathbb{P}^1 \setminus \left\{0, 1, \infty\right\} \setminus \left\{\mathrm{diagonals}\right\}\right)/S_3.$

Observe that the moduli space is three-dimensional, as predicted by a deformation theoretic calculation that identifies the tangent space to the moduli space (or rather, the moduli stack) at a curve ${C}$ with ${H^1(T_C)}$.

A striking feature here is that the moduli space ${M_2}$ is unirational: that is, it admits a dominant rational map from a projective space. In fact, one even has a little more: one has a family of genus curves over an open subset in projective space (given by the family ${y^2 = \prod (x - x_i)}$ as the ${\left\{x_i\right\}}$ as vary) such that every genus two curve occurs in the family (albeit more than once).

The simplicity of ${M_2}$, and in particular the parametrization of genus two curves by points in a projective space, is a low genus phenomenon, although similar “classifications” can be made in a few higher genera. (For example, a general genus four curve is an intersection of a quadric and cubic in ${\mathbb{P}^3}$, and one can thus parametrize most genus four curves by a rational variety.) As ${g \rightarrow \infty}$, the variety ${M_g}$ parametrizing genus ${g}$ curves is known to be of general type, by a theorem of Harris and Mumford. (more…)

Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) ${X}$. One of the first facts one observes in their theory is that the group ${\mathrm{Aut}(X)}$ of automorphisms of ${X}$ is quite large when ${X}$ has genus zero or one. When ${X}$ has genus zero, it is the projective line, and its automorphism group is ${\mathrm{PGL}_2(\mathbb{C})}$, a fact which generalizes naturally to higher projective spaces. When ${g = 1}$, the curve ${X}$ acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of ${X}$ act on ${X}$. So ${\mathrm{Aut}(X)}$ has points corresponding to each element of ${X}$ (and a few more, such as inversion in the group law).

But when the genus of ${X}$ is at least two, things change dramatically. It is a famous theorem that the number of automorphisms is bounded:

Theorem 1 (Hurwitz) If ${X}$ is a compact Riemann surface of genus ${g \geq 2}$, then there are at most ${84(g-1)}$ automorphisms of ${X}$.

I won’t write out a proof here; a discussion is in lecture 9 of the notes I’m taking in an algebraic curves class. In fact, the hard part is to show that there are finitely many automorphisms, after which it is a combinatorial argument.

This bound is often sharp. There are infinitely many genera ${g}$ together with Riemann surfaces with exactly ${84(g-1)}$ automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic $p$, because the Riemann-Hurwitz formula is no longer necessarily true (because of the existence of non-separable morphisms), and counterexamples are given in the notes.

What I want to describe today is that this bound is often not sharp.

Theorem 2 If ${X}$ is a compact Riemann surface of genus ${g = p+1}$ for ${p>84}$ a prime, then ${\mathrm{Aut}(X)}$ has order strictly less than ${84(g-1)}$. (more…)

I’ve been live-TeXing a course of Joe Harris, titled “Geometry of Algebraic Curves,” a second course in the theory of algebraic curves. Here are the notes, which are updated after each lecture.