# Hyper-elliptic curve

2010 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.

#### References

 [1] C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , Amer. Math. Soc. (1951) MR0042164 Zbl 0045.32301 [2] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602 [3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001