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Riemann–Roch theorem for surfaces
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 , after preliminary versions of it were found by and . The sheaf-theoretic version is due to Hirzebruch.
==Statement==
One form of the Riemann–Roch theorem states that if ''D'' is a divisor on a non-singular projective surface then
:
where χ is the 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''''a'', where ''p''''a'' 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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