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 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 four 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.
2. The Jacobian
The Jacobian of a genus two curve can also be described (somewhat) explicitly. Namely, one knows that, for any genus curve
, the Jacobian
is birational to the symmetric power
, and is the quotient of that by linear equivalence.
For , we have a smooth surface
, which is also the Hilbert scheme of length two subschemes on
: that is, it parametrizes degree two effective divisors on
. The degree two (canonical) map
has the property that its fibers form a ‘s worth of linearly equivalent degree two divisors. But this is the only linear equivalence that occurs: if
is any degree two divisor with
, then
by Riemann-Roch. It follows that the Jacobian
is obtained from
by contracting — by blowing down — the
of divisors in the canonical series.
For a general genus two curve , the Jacobian
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
), the Jacobian
will be non-simple, or equivalently there will exist an isogeny
for two elliptic curves . The curves
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 is isogeneous to a product of elliptic curves, then there exists a surjection
for an elliptic curve ; this forces the existence of a nonconstant map
. Conversely, a nonconstant map
would lead to a surjection
and thus a decomposition of
. 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 from a genus two curve to a genus one curve
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 that admit a degree two map to an elliptic curve.
Example (Jacobi): Let be an involution; it has two fixed points
. We can move these to
, respectively, and thus assume that the involution is given by multiplication by
.
Given three nonzero complex numbers , we consider the genus two curve
given by
This has two natural involutions. First, there is the hyperelliptic involution . But second, there is the involution
.
Let’s consider the quotient of by the second involution. We have a diagram
where the vertical maps are degree two. Note that is ramified at the fixed points of
, which are precisely the points of
lying above
. (The points lying above
are permuted: the involution
interchanges the two “asymptotes” of
.) Thus there are two branch points at
, which by Riemann-Hurwitz implies that
has genus one.
So this is a construction of genus two curves with split Jacobian, starting from three distinct points . The associated elliptic curve
comes with a degree two map to
, which is branched over the images of
(since
is) as well as above
.
June 20, 2013 at 8:44 am
What about positive characteristic?
June 21, 2013 at 9:23 pm
I believe everything in the above post works away from characteristic 2. I’m not really clear on the analog of the equation
in characteristic two; I’d imagine it is some sort of Artin-Schreier equation that is relevant. (Of course, you know all this much better than I!)