Editing Hyperelliptic curve (section)

==Formulation and choice of model==

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a [[Mathematical singularity|singular point]] ''at infinity'' in the [[projective plane]]. This feature is specific to the case ''n'' > 3. 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 an open cover of the curve by two affine charts: the one already given by
<math display="block">y^2 = f(x) </math>
and another one given by
<math display="block">w^2 = v^{2g+2}f(1/v) .</math>

The glueing maps between the two charts are given by
<math display="block">(x,y) \mapsto (1/x, y/x^{g+1})</math>
and
<math display="block">(v,w) \mapsto (1/v, w/v^{g+1}),</math>
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 (mathematics)|ramification]] occurring at the roots of ''f'', and also for odd ''n'' at the point at infinity. In this way the cases ''n'' = 2''g'' + 1 and 2''g'' + 2 can be unified, since we might as well use an [[automorphism]] of the projective plane to move any ramification point away from infinity.