Editing Canonical bundle (section)

====Hyperelliptic case====

If ''C'' has genus two or more, then the canonical class is [[big line bundle|big]], so the image of any ''n''-canonical map is a curve. The image of the 1-canonical map is called a '''[[canonical curve]]'''. A canonical curve of genus ''g'' always sits in a projective space of dimension {{nowrap begin}}''g'' &minus; 1{{nowrap end}}.<ref name = Parshin>{{springer| title= Canonical curve | id= c/c020150 | last= Parshin | first= A. N.}}</ref> When ''C'' is a [[hyperelliptic curve]], the canonical curve is a [[rational normal curve]], and ''C'' a double cover of its canonical curve. For example if ''P'' is a polynomial of degree 6 (without repeated roots) then

:''y''<sup>2</sup> = ''P''(''x'')

is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by

:''dx''/{{radic|''P''(''x'')}}, &nbsp; ''x dx''/{{radic|''P''(''x'')}}.

This means that the canonical map is given by [[homogeneous coordinates]] [1: ''x''] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in ''x''.