View source for Hyper-elliptic curve

A non-singular projective model of the affine curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482102.png" /> is a polynomial without multiple roots of odd degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482103.png" /> (the case of even degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482104.png" /> may be reduced to that of odd degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482105.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482106.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482107.png" />, so that, for various odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482108.png" />, hyper-elliptic curves are birationally inequivalent. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h0482109.png" /> one obtains the projective straight line; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h04821010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048210/h04821011.png" />; this property is a complete characterization of hyper-elliptic curves.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , Amer. Math. Soc. (1951) {{MR|0042164}} {{ZBL|0045.32301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>



====Comments====
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|Covering surface]]) of a [[Rational curve|rational curve]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1985) {{MR|0770932}} {{ZBL|0559.14017}} </TD></TR></table>