Editing Canonical bundle (section)

====General case====

Otherwise, for non-hyperelliptic ''C'' which means ''g'' is at least 3, the morphism is an isomorphism of ''C'' with its image, which has degree 2''g'' &minus; 2. Thus for ''g'' = 3 the canonical curves (non-hyperelliptic case) are [[quartic plane curve]]s. All non-singular plane quartics arise in this way. There is explicit information for the case ''g'' = 4, when a canonical curve is an intersection of a [[quadric]] and a [[cubic surface]]; and for ''g'' = 5 when it is an intersection of three quadrics.<ref name = Parshin/> There is a converse, which is a corollary to the [[Riemann–Roch theorem]]: a non-singular curve ''C'' of genus ''g'' embedded in projective space of dimension ''g'' &minus; 1 as a [[linearly normal]] curve of degree 2''g'' &minus; 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves ''C''  (in the non-hyperelliptic case of ''g'' at least 3), Riemann-Roch, and the theory of [[special divisor]]s is rather close. Effective divisors ''D'' on ''C'' consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.<ref>{{cite web |url=http://rigtriv.wordpress.com/2008/08/07/geometric-form-of-riemann-roch/ |title = Geometric Form of Riemann-Roch {{!}} Rigorous Trivialities| date=7 August 2008 }}</ref><ref>Rick Miranda, ''Algebraic Curves and Riemann Surfaces'' (1995), Ch. VII.</ref>

More refined information is available, for larger values of ''g'', but in these cases canonical curves are not generally [[complete intersection]]s, and the description requires more consideration of [[commutative algebra]]. The field started with '''Max Noether's theorem''': the dimension of the space of quadrics passing through ''C'' as embedded as canonical curve is (''g'' &minus; 2)(''g'' &minus; 3)/2.<ref>[[David Eisenbud]], ''The Geometry of Syzygies'' (2005), p. 181-2.</ref> '''Petri's theorem''', often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for ''g'' at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) [[trigonal curve]]s and (b) non-singular plane quintics when ''g'' = 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, [[Oscar Chisini]] and [[Federigo Enriques]]). The terminology is confused, since the result is also called the '''Noether–Enriques theorem'''. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is [[normally generated]]: the [[symmetric power]]s of the space of sections of the canonical bundle map onto the sections of its tensor powers.<ref>{{springer| title= Noether–Enriques theorem | id= N/n066770 | last= Iskovskih | first= V. A.}}</ref><ref>[[Igor Rostislavovich Shafarevich]], ''Algebraic geometry I'' (1994), p. 192.</ref> This implies for instance the generation of the [[quadratic differential]]s on such curves by the differentials of the first kind; and this has consequences for the [[local Torelli theorem]].<ref>{{Springer|title=Torelli theorems|id=T/t093260}}</ref> Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a [[ruled surface]] and a [[Veronese surface]].

These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.<ref>http://hal.archives-ouvertes.fr/docs/00/40/42/57/PDF/these-OD.pdf, pp. 11-13.</ref>