Hyper-elliptic curve

2020 Mathematics Subject Classification: Primary: 14H45 [MSN][ZBL]

A non-singular projective model of the affine curve $y^2=f(x)$, where $f(x)$ is a polynomial without multiple roots of odd degree $n$ (the case of even degree $2k$ may be reduced to that of odd degree $2k-1$). The field of functions on a hyper-elliptic curve (a field of hyper-elliptic functions) is a quadratic extension of the field of rational functions; in this sense it is the simplest field of algebraic functions except for the field of rational functions. Hyper-elliptic curves are distinguished by the condition of the existence of a one-dimensional linear series $g_2'$ of divisors of degree 2, defining a morphism of order 2 of the hyper-elliptic curve onto the projective straight line. The genus of a hyper-elliptic curve is $g =(n-1)/2$, so that, for various odd $n$, hyper-elliptic curves are birationally inequivalent.

For $n=1$, $g=0$ one obtains the projective straight line; for $n=3$, $g=1$ an elliptic curve is obtained. Traditionally, curves of genus 0 and 1 are not called hyper-elliptic curves. The fractions of regular differential forms generate a subfield of genus 0 on a hyper-elliptic curve of genus $g>1$; this property is a complete characterization of hyper-elliptic curves. A further characterization is that hyper-elliptic curves have exactly $2g+2$ Weierstrass points.

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The definition given in the main article (first sentence) is only valid in characteristic not equal to 2. In general, a hyper-elliptic curve can be defined as a double covering (cf. also Covering surface) of a rational curve.

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