Hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve given by an equation of the form
where f(x) is a polynomial of degree n > 4 with n distinct roots. A hyperelliptic function is an element of the function field of such a curve or possibly of the Jacobian variety on the curve, these two concepts being the same in the elliptic function case, but different in the present case. Fig. 1 is the graph of where
The degree of the polynomial determines the genus of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve. Meanwhile, a curve of degree 2g + 2 is termed a real hyperelliptic curve. This statement about genus remains true for g = 0 or 1, but those curves are not called "hyperelliptic". Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve. Hence the terminology.
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.
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