View source for Riemann–Roch theorem for surfaces

{{Short description|Mathematical theorem}}
{{Infobox mathematical statement
| name = Riemann–Roch theorem for surfaces
| image = 
| caption = 
| field = [[Algebraic geometry]]
| conjectured by = 
| conjecture date = 
| first proof by = [[Biography:Guido Castelnuovo|Guido Castelnuovo]], [[Biography:Max Noether|Max Noether]], [[Biography:Federigo Enriques|Federigo Enriques]]
| first proof date = 1886, 1894, 1896, 1897
| open problem = 
| known cases = 
| implied by =
| equivalent to = 
| generalizations = [[Atiyah–Singer index theorem]]<br>[[Grothendieck–Riemann–Roch theorem]]<br>[[Hirzebruch–Riemann–Roch theorem]]
| consequences = [[Riemann–Roch theorem]]
}}

In mathematics, the '''Riemann–Roch theorem for surfaces''' describes the dimension of linear systems on an [[Algebraic surface|algebraic surface]]. The classical form of it was first given by {{harvs|txt|last=Castelnuovo|year1=1896|year2=1897}}, after preliminary versions of it were found by {{harvs|txt|first=Max|last=Noether|authorlink=Max Noether|year=1886}} and {{harvs|txt|last=Enriques|authorlink=Federigo Enriques|year=1894}}. The [[Sheaf (mathematics)|sheaf]]-theoretic version is due to Hirzebruch.

==Statement==
One form of the Riemann&ndash;Roch theorem states that if ''D'' is a divisor on a non-singular projective surface then
:<math>\chi(D) = \chi(0) +\tfrac{1}{2} D . (D - K) \,</math>

where &chi; is the holomorphic Euler characteristic, the dot . is the [[Intersection number|intersection number]], and ''K'' is the canonical divisor. The constant &chi;(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1&nbsp;+&nbsp;''p''<sub>''a''</sub>, where ''p''<sub>''a''</sub> is the [[Arithmetic genus|arithmetic genus]] of the surface. For comparison, the Riemann&ndash;Roch theorem for a curve states that&nbsp;&chi;(''D'')&nbsp;=&nbsp;&chi;(0)&nbsp;+&nbsp;deg(''D'').

==Noether's formula==

[[Biography:Max Noether|Noether's]] formula states that

:<math>\chi = \frac{c_1^2+c_2}{12} = \frac{(K.K)+e}{12}</math>

where &chi;=&chi;(0) is the holomorphic Euler characteristic, ''c''<sub>1</sub><sup>2</sup> =&nbsp;(''K''.''K'') is a [[Chern class#Chern number|Chern number]] and the self-intersection number of the canonical class ''K'', and ''e''&nbsp;=&nbsp;''c''<sub>2</sub> is the topological Euler characteristic. It can be used to replace the 
term &chi;(0) in the Riemann–Roch theorem with topological terms; this gives the [[Hirzebruch–Riemann–Roch theorem]] for surfaces.

==Relation to the Hirzebruch–Riemann–Roch theorem==
For surfaces, the [[Hirzebruch–Riemann–Roch theorem]] is essentially the Riemann–Roch theorem for surfaces combined with the Noether formula. To see this, recall that for each divisor ''D'' on a surface there is an [[Invertible sheaf|invertible sheaf]] ''L'' = O(''D'') such that the linear system of ''D'' is more or less the space of sections of ''L''. 
For surfaces the Todd class is <math>1 + c_1(X) / 2 + (c_1(X)^2 + c_2(X)) / 12</math>, and the Chern character of the sheaf ''L'' is just <math>1 + c_1(L) + c_1(L)^2 / 2</math>, so the Hirzebruch–Riemann–Roch theorem states that

: <math>
\begin{align}
\chi(D) &= h^0(L) - h^1(L) + h^2(L)\\
&= \frac{1}{2} c_1(L)^2 + \frac{1}{2} c_1(L) \, c_1(X) + \frac{1}{12} \left(c_1(X)^2 + c_2(X)\right)
\end{align}
</math>

Fortunately this can be written in a clearer form as follows. First putting ''D''&nbsp;=&nbsp;0 shows that

: <math> \chi(0) = \frac{1}{12}\left(c_1(X)^2 + c_2(X)\right)</math> &nbsp; &nbsp; (Noether's formula)

For invertible sheaves (line bundles) the second Chern class vanishes. The products of second cohomology classes can be identified with intersection numbers in the [[Picard group]], and we get a more classical version of Riemann Roch for surfaces:

: <math> \chi(D) = \chi(0) + \frac{1}{2}(D.D - D.K) </math>

If we want, we can use [[Serre duality]] to express ''h''<sup>2</sup>(O(''D'')) as ''h''<sup>0</sup>(O(''K''&nbsp;&minus;&nbsp;''D'')), but unlike the case of curves there is in general no easy way to write the ''h''<sup>1</sup>(O(''D'')) term in a form not involving sheaf cohomology (although in practice it often vanishes).

==Early versions==
The earliest forms of the Riemann&ndash;Roch theorem for surfaces were often stated as an inequality rather than an equality, because there was no direct geometric description of first cohomology groups. 
A typical example is given by {{harvtxt|Zariski|1995|loc=p.&nbsp;78}}, which states that

:<math>r\ge n-\pi+p_a+1-i</math>

where
*''r'' is the dimension of the complete linear system |''D''| of a divisor ''D'' (so ''r''&nbsp;=&nbsp;''h''<sup>0</sup>(O(''D'')) &minus;1)
*''n'' is the '''virtual degree''' of ''D'', given by the self-intersection number (''D''.''D'')
*&pi; is the '''virtual genus''' of ''D'', equal to 1 + (D.D + K.D)/2
*''p''<sub>''a''</sub> is the '''arithmetic genus''' &chi;(O<sub>''F''</sub>) &minus; 1 of the surface
*''i'' is the '''index of speciality''' of ''D'', equal to dim ''H''<sup>0</sup>(O(''K''&nbsp;&minus;&nbsp;''D'')) (which by Serre duality is the same as dim ''H''<sup>2</sup>(O(D))).

The difference between the two sides of this inequality was called the '''superabundance''' ''s''  of the divisor ''D''. 
Comparing this inequality with the sheaf-theoretic version of the Riemann&ndash;Roch theorem shows that the superabundance of ''D'' is given by ''s''&nbsp;=&nbsp;dim ''H''<sup>1</sup>(O(''D'')). The divisor ''D'' was called '''regular''' if ''i''&nbsp;=&nbsp;''s''&nbsp;=&nbsp;0 (or in other words if all higher cohomology groups of O(''D'') vanish)  and '''superabundant''' if&nbsp;''s''&nbsp;>&nbsp;0.

==References==
* ''Topological Methods in Algebraic Geometry'' by Friedrich Hirzebruch {{isbn|3-540-58663-6}}
*{{Citation | last1=Zariski | first1=Oscar |  title=Algebraic surfaces | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58658-6 |mr=1336146 | year=1995}}
*{{cite web |last1=Smith |first1=Roy |title=On Classical Riemann Roch and Hirzebruch's generalization |url=https://www.math.uga.edu/sites/default/files/inline-files/rrt.pdf |website=Department of Mathematics Boyd Research and Education Center University of Georgia}}
{{DEFAULTSORT:Riemann-Roch theorem for surfaces}}
[[Category:Theorems in algebraic geometry]]
[[Category:Algebraic surfaces]]
[[Category:Topological methods of algebraic geometry]]

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