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.

2. The Jacobian

The Jacobian of a genus two curve can also be described (somewhat) explicitly. Namely, one knows that, for any genus ${g}$ curve ${C}$, the Jacobian ${J(C)}$ is birational to the symmetric power ${C_g = \mathrm{Sym}^g C}$, and is the quotient of that by linear equivalence.

For ${g = 2}$, we have a smooth surface ${\mathrm{Sym}^2 C}$, which is also the Hilbert scheme of length two subschemes on ${C}$: that is, it parametrizes degree two effective divisors on ${C}$. The degree two (canonical) map

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

has the property that its fibers form a ${\mathbb{P}^1}$‘s worth of linearly equivalent degree two divisors. But this is the only linear equivalence that occurs: if ${D}$ is any degree two divisor with ${H^0(\mathcal{O}(D)) \geq 2}$, then ${D \sim K}$ by Riemann-Roch. It follows that the Jacobian ${J(C)}$ is obtained from ${\mathrm{Sym}^2 C}$ by contracting — by blowing down — the ${\mathbb{P}^1}$ of divisors in the canonical series.

For a general genus two curve ${C}$, the Jacobian ${J(C)}$ will be a simple abelian surface: it will not admit any nontrivial abelian subvarieties. However, for some (precisely, for a union of countably many divisors in ${M_2}$), the Jacobian ${J(C)}$ will be non-simple, or equivalently there will exist an isogeny

$\displaystyle J(C) \sim E_1 \times E_2,$

for two elliptic curves ${E_1, E_2}$. The curves ${E_1, E_2}$ are determined uniquely up to isogeny by the corollary of the Poincaré complete reducibility theorem that states that abelian varieties up to isogeny form a semisimple abelian category.

If ${J(C)}$ is isogeneous to a product of elliptic curves, then there exists a surjection

$\displaystyle J(C) \twoheadrightarrow E,$

for an elliptic curve ${E}$; this forces the existence of a nonconstant map ${C \rightarrow J(C) \rightarrow E}$. Conversely, a nonconstant map ${C \rightarrow E}$ would lead to a surjection ${J(C) \rightarrow J(E) \simeq E}$ and thus a decomposition of ${J(C)}$. It follows that the genus two curves whose Jacobian decomposes in this way are precisely those which admit map to an elliptic curve.

The Riemann-Hurwitz theorem does not rule out a map ${C \rightarrow E}$ from a genus two curve to a genus one curve ${E}$ of anydegree, provided there are two branch points. To give a genus two curve which maps to an elliptic curve, it follows that one must give an elliptic curve with one other additional marked point (in addition to the origin), together with some discrete combinatorial (monodromy) data; the family of such is two-dimensional. This justifies the claim that there is a two-dimensional family of genus two curves with non-simple Jacobian.

It is possible to completely write down these curves ${C}$ that admit a degree two map to an elliptic curve.

Example (Jacobi): Let ${\iota: \mathbb{P}^1 \rightarrow \mathbb{P}^1}$ be an involution; it has two fixed points ${p, q \in \mathbb{P} ^1}$. We can move these to ${\left\{0, \infty\right\}}$, respectively, and thus assume that the involution is given by multiplication by ${-1}$.

Given three nonzero complex numbers ${x_1, x_2, x_3}$, we consider the genus two curve ${C}$ given by

$\displaystyle y^2 = \prod_1^3 (x - x_i)(x + x_i).$

This has two natural involutions. First, there is the hyperelliptic involution ${(x,y) \mapsto (x, -y)}$. But second, there is the involution ${I: (x, y) \mapsto (- x, y)}$.

Let’s consider the quotient of ${C}$ by the second involution. We have a diagram

where the vertical maps are degree two. Note that ${C \rightarrow C/I}$ is ramified at the fixed points of ${I}$, which are precisely the points of ${C}$ lying above ${x = 0}$. (The points lying above ${x = \infty}$ are permuted: the involution ${I}$ interchanges the two “asymptotes” of ${C}$.) Thus there are two branch points at ${C \rightarrow C/I}$, which by Riemann-Hurwitz implies that ${C/I}$ has genus one.

So this is a construction of genus two curves with split Jacobian, starting from three distinct points ${x_1, x_2, x_3 \in \mathbb{C}^{*}}$. The associated elliptic curve ${C/I}$ comes with a degree two map to ${\mathbb{P}^1/\iota \simeq \mathbb{P}^1}$, which is branched over the images of ${\left\{x_1, x_2, x_3\right\}}$ (since ${C \rightarrow \mathbb{P}^1}$ is) as well as above ${\infty}$.