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| In mathematics, the '''Riemann–Roch theorem for surfaces''' describes the dimension of linear systems on an [[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 {{harvtxt|Noether|1886}} and {{harvtxt|Enriques|1894}}. The [[sheaf (mathematics)|sheaf]]-theoretic version is due to Hirzebruch.
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| ==Statement==
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| One form of the Riemann–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>
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| where χ is the [[sheaf theory|holomorphic Euler characteristic]], the dot . is the [[intersection number]], and ''K'' is the canonical divisor. The constant χ(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + ''p''<sub>''a''</sub>, where ''p''<sub>''a''</sub> is the [[arithmetic genus]] of the surface. For comparison, the Riemann–Roch theorem for a curve states that χ(''D'') = χ(0) + deg(''D'').
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| ==Noether's formula==
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| [[Max Noether|Noether's]] formula states that
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| :<math>\chi = \frac{c_1^2+c_2}{12} = \frac{(K.K)+e}{12}</math>
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| where χ=χ(0) is the holomorphic Euler characteristic, ''c''<sub>1</sub><sup>2</sup> = (''K''.''K'') is a [[Chern class#Chern number|Chern number]] and the self-intersection number of the canonical class ''K'', and ''e'' = ''c''<sub>2</sub> is the topological Euler characteristic. It can be used to replace the
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| term χ(0) in the Riemann–Roch theorem with topological terms; this gives the [[Hirzebruch–Riemann–Roch theorem]] for surfaces.
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| ==Relation to the Hirzebruch–Riemann–Roch theorem==
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| 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]] ''L'' = O(''D'') such that the linear system of ''D'' is more or less the space of sections of ''L''.
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| 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
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| : <math>
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| \begin{align}
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| \chi(D) &= h^0(L) - h^1(L) + h^2(L)\\
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| &= \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)
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| \end{align}
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| </math>
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| Fortunately this can be written in a clearer form as follows. First putting ''D'' = 0 shows that
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| : <math> \chi(0) = \frac{1}{12}\left(c_1(X)^2 + c_2(X)\right)</math> (Noether's formula)
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| 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:
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| : <math> \chi(D) = \chi(0) + \frac{1}{2}(D.D - D.K) </math>
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| If we want, we can use [[Serre duality]] to express ''h''<sup>2</sup>(O(''D'')) as ''h''<sup>0</sup>(O(''K'' − ''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).
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| ==Early versions==
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| The earliest forms of the Riemann–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.
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| A typical example is given by {{harvtxt|Zariski|1995|loc=p. 78}}, which states that
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| :<math>r\ge n-\pi+p_a+1-i</math> | |
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| where
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| *''r'' is the dimension of the complete linear system |''D''| of a divisor ''D'' (so ''r'' = ''h''<sup>0</sup>(O(''D'')) −1)
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| *''n'' is the '''virtual degree''' of ''D'', given by the self-intersection number (''D''.''D'')
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| *π is the '''virtual genus''' of ''D'', equal to 1 + (D.D + K)/2
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| *''p''<sub>''a''</sub> is the '''arithmetic genus''' χ(O<sub>''F''</sub>) − 1 of the surface
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| *''i'' is the '''index of speciality''' of ''D'', equal to dim ''H''<sup>0</sup>(O(''K'' − ''D'')) (which by Serre duality is the same as dim ''H''<sup>2</sup>(O(D))).
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| The difference between the two sides of this inequality was called the '''superabundance''' ''s'' of the divisor ''D''.
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| Comparing this inequality with the sheaf-theoretic version of the Riemann–Roch theorem shows that the superabundance of ''D'' is given by ''s'' = dim ''H''<sup>1</sup>(O(''D'')). The divisor ''D'' was called '''regular''' if ''i'' = ''s'' = 0 (or in other words if all higher cohomology groups of O(''D'') vanish) and '''superabundant''' if ''s'' > 0.
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| ==References==
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| * ''Topological Methods in Algebraic Geometry'' by Friedrich Hirzebruch ISBN 3-540-58663-6
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| *{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58658-6 | id={{MathSciNet | id = 1336146}} | year=1995}}
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| {{DEFAULTSORT:Riemann-Roch theorem for surfaces}}
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| [[Category:Theorems in algebraic geometry]]
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| [[Category:Algebraic surfaces]]
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| [[Category:Topological methods of algebraic geometry]]
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