Editing Riemann–Roch theorem (section)

== Statement of the theorem ==

The Riemann–Roch theorem for a compact Riemann surface of genus <math>g</math> with canonical divisor <math>K</math> states

:<math>\ell(D)-\ell(K-D)=\deg(D)-g+1</math>.

Typically, the number <math>\ell(D)</math> is the one of interest, while <math>\ell(K-D)</math> is thought of as a correction term (also called index of speciality<ref>Stichtenoth p.22</ref><ref>Mukai pp.295–297</ref>) so the theorem may be roughly paraphrased by saying

:''dimension'' − ''correction'' = ''degree'' − ''genus'' + 1.

Because it is the dimension of a vector space, the correction term <math>\ell(K-D)</math> is always non-negative, so that

:<math>\ell(D)\ge\deg(D)-g+1</math>.

This is called ''Riemann's inequality''. ''Roch's part'' of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus <math>g</math>, <math>K</math> has degree <math>2g-2</math>, independently of the meromorphic form chosen to represent the divisor. This follows from putting <math>D=K</math> in the theorem. In particular, as long as <math>D</math> has degree at least <math>2g-1</math>, the correction term is 0, so that

:<math>\ell(D)=\deg(D)-g+1</math>.

The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using [[line bundle]]s and a generalization of the theorem to [[algebraic curve]]s.

===Examples===
The theorem will be illustrated by picking a point <math>P</math> on the surface in question and regarding the sequence of numbers

:<math>\ell(n\cdot P),n\ge0</math>

i.e., the dimension of the space of functions that are holomorphic everywhere except at <math>P</math> where the function is allowed to have a pole of order at most <math>n</math>. For <math>n=0</math>, the functions are thus required to be [[entire function|entire]], i.e., holomorphic on the whole surface <math>X</math>. By [[Liouville's theorem (complex analysis)#On compact Riemann surfaces|Liouville's theorem]], such a function is necessarily constant. Therefore, <math>\ell(0)=1</math>. In general, the sequence <math>\ell(n\cdot P)</math> is an increasing sequence.

====Genus zero====
The [[Riemann sphere]] (also called [[complex projective line]]) is [[simply connected]] and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of <math>\Complex</math>, with [[transition map]] being given by

:<math>\Complex\setminus\{0\}\ni z\mapsto\frac{1}{z}\in\Complex\setminus\{0\}</math>.

Therefore, the form <math>\omega = dz</math> on one copy of <math>\mathbb C</math> extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since

:<math>d\left(\frac1z\right)=-\frac1{z^2}\,dz</math>

Thus, its canonical divisor is <math>K:=\operatorname{div}(\omega)=-2P</math> (where <math>P</math> is the point at infinity).

Therefore, the theorem says that the sequence <math>\ell(n\cdot P)</math> reads

: 1, 2, 3, ... .

This sequence can also be read off from the theory of [[partial fraction]]s. Conversely if this sequence starts this way, then <math>g</math> must be zero.

====Genus one====
[[File:Torus_cycles2.svg|right|thumb|A torus]]
The next case is a Riemann surface of genus <math>g=1</math>, such as a [[torus]] <math>\Complex/\Lambda</math>, where <math>\Lambda</math> is a two-dimensional [[lattice (group)|lattice]] (a group isomorphic to <math>\Z^2</math>). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate <math>z</math> on <math>C</math> yields a one-form <math>\omega=dz</math> on <math>X</math> that is everywhere holomorphic, i.e., has no poles at all. Therefore, <math>K</math>, the divisor of <math>\omega</math> is zero.

On this surface, this sequence is

:1, 1, 2, 3, 4, 5 ... ;

and this characterises the case <math>g=1</math>. Indeed, for <math>D=0</math>, <math>\ell(K-D)=\ell(0)=1</math>, as was mentioned above. For <math>D=n\cdot P</math> with <math>n>0</math>, the degree of <math>K-D</math> is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of [[elliptic function]]s.

====Genus two and beyond====
For <math>g=2</math>, the sequence mentioned above is

:1, 1, ?, 2, 3, ... .

It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a [[hyperelliptic curve]]. For <math>g>2</math> it is always true that at most points the sequence starts with <math>g+1</math> ones and there are finitely many points with other sequences (see [[Weierstrass point]]s).

===Riemann–Roch for line bundles===
Using the close correspondence between divisors and [[holomorphic line bundle]]s on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let ''L'' be a holomorphic line bundle on ''X''. Let <math>H^0(X,L)</math> denote the space of holomorphic sections of ''L''. This space will be finite-dimensional; its dimension is denoted <math>h^0(X,L)</math>. Let ''K'' denote the [[canonical bundle]] on ''X''. Then, the Riemann–Roch theorem states that

:<math>h^0(X,L)-h^0(X,L^{-1}\otimes K)=\deg(L)+1-g</math>.

The theorem of the previous section is the special case of when ''L'' is a [[point bundle]].

The theorem can be applied to show that there are ''g'' linearly independent holomorphic sections of ''K'', or [[one-form]]s on ''X'', as follows. Taking ''L'' to be the trivial bundle, <math> h^0(X,L)=1</math> since the only holomorphic functions on ''X'' are constants. The degree of ''L'' is zero, and <math>L^{-1}</math> is the trivial bundle. Thus,

:<math>1-h^0(X,K)=1-g</math>.

Therefore, <math>h^0(X,K)=g</math>, proving that there are ''g'' holomorphic one-forms.

=== Degree of canonical bundle ===
Since the canonical bundle <math>K</math> has <math>h^0(X,K)=g</math>, applying Riemann–Roch to <math>L=K</math> gives

:<math>h^0(X,K)-h^0(X,K^{-1}\otimes K)=\deg(K)+1-g</math>

which can be rewritten as

:<math>g-1=\deg(K)+1-g</math>

hence the degree of the canonical bundle is <math>\deg(K)=2g-2</math>.

===Riemann–Roch theorem for algebraic curves===
Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in [[algebraic geometry]]. The analogue of a Riemann surface is a [[Singular point of an algebraic variety|non-singular]] [[algebraic curve]] ''C'' over a field ''k''. The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a real [[manifold]] is two, but one as a complex manifold. The compactness of a Riemann surface is paralleled by the condition that the algebraic curve be [[complete variety|complete]], which is equivalent to being [[projective variety|projective]]. Over a general field ''k'', there is no good notion of singular (co)homology. The so-called [[geometric genus]] is defined as

:<math>g(C):=\dim_k\Gamma(C,\Omega^1_C)</math>

i.e., as the dimension of the space of globally defined (algebraic) one-forms (see [[Kähler differential]]). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced by [[rational function]]s which are locally fractions of [[regular function]]s. Thus, writing <math>\ell(D)</math> for the dimension (over ''k'') of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient in ''D'', the very same formula as above holds:

:<math>\ell(D)-\ell(K-D)=\deg(D)-g+1</math>.

where ''C'' is a projective non-singular algebraic curve over an [[algebraically closed field]] ''k''. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account [[multiplicity (mathematics)|multiplicities]] coming from the possible extensions of the base field and the [[residue field]]s of the points supporting the divisor.<ref>{{Citation | last1=Liu | first1=Qing | title=Algebraic Geometry and Arithmetic Curves | publisher=[[Oxford University Press]] | isbn=978-0-19-850284-5 | year=2002}}, Section 7.3</ref> Finally, for a proper curve over an [[Artinian ring]], the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf <math>\mathcal O</math>.<ref>* {{Citation | last1=Altman | first1=Allen | last2=Kleiman | first2=Steven | author2-link=Steven Kleiman | title=Introduction to Grothendieck duality theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 146 | year=1970}}, Theorem VIII.1.4., p. 164</ref>

The smoothness assumption in the theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are [[Gorenstein ring]]s, the same statement as above holds, provided that the geometric genus as defined above is replaced by the [[arithmetic genus]] ''g''<sub>''a''</sub>, defined as

:<math>g_a:=\dim_k H^1(C,\mathcal O_C)</math>.<ref>{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Generalized divisors on Gorenstein curves and a theorem of Noether | year=1986 | journal=Journal of Mathematics of Kyoto University | issn=0023-608X | volume=26 | issue=3 | pages=375–386 | doi=10.1215/kjm/1250520873 | doi-access=free }}</ref>

(For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties).<ref>{{Citation | last1=Baum | first1=Paul | last2=Fulton | first2=William | author2-link=William Fulton (mathematician) | last3=MacPherson | first3=Robert | author3-link=Robert MacPherson (mathematician) | title=Riemann–Roch for singular varieties | year=1975 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=45 | issn=1618-1913 | issue=45 | pages=101–145| doi=10.1007/BF02684299 | s2cid=83458307 | url=http://www.numdam.org/item/PMIHES_1975__45__101_0/ }}</ref>