Hyperelliptic Curve - Formulation and Choice of Model

Formulation and Choice of Model

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular point at infinity in the projective plane. This feature is specific to the case n > 4. Therefore in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant.

To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is a cover of the curve with two affine pieces: the one already given by

and another one given by

.

The glueing maps between the two pieces are given by

and

wherever they are defined.

In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at the point at infinity. In this way the cases n = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism of the projective line to move any ramification point away from infinity.