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…)