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.
Every genus two curve is hyperelliptic in a natural manner. As with any curve, the canonical line bundle is generated by global sections. Since there are two linearly independent holomorphic differentials on , one gets a map
Since has degree two, the map is a two-fold cover: that is, is a hyperelliptic curve. In particular, as with any two-fold cover, there is a canonical involution of the cover , 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 curve has no automorphisms.
A count using Riemann-Hurwitz shows that the canonical map must be branched at precisely six points, which we can assume are . There is no further monodromy data to give for the cover , since it is a two-fold cover; it follows that is exhibited as the Riemann surface associated to the equation
More precisely, the curve is cut out in weighted projective space by the homogenized form of the above equation,
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 , modulo automorphisms of . In other words, one takes an open subset , and quotients by the action of . In fact, this is a description of the coarse moduli space of genus two curves: that is, it is a variety whose complex points parametrize precisely genus two curves, and which is “topologized” such that any family of genus two curves over a base gives a map . Moreover, is initial with respect to this property.
It can sometimes simplify things to assume that three of the branch points in are given by , which rigidifies most of the action of ; then one simply has to choose three (unordered) distinct points on modulo action of the group consisting of automorphisms of that preserve . In other words,
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 with .
A striking feature here is that the moduli space 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 as the as vary) such that every genus two curve occurs in the family (albeit more than once).
The simplicity of , 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 , and one can thus parametrize most genus three curves by a rational variety.) As , the variety parametrizing genus curves is known to be of general type, by a theorem of Harris and Mumford. (more…)