Castelnuovo-de Franchis theorem

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In mathematics, the Castelnuovo-de Franchis theorem is a classical result on complex algebraic surfaces. If X is such a surface, projective and non-singular, suppose given two differentials of the first kind

ω_i with i = 1,2

on X which are linearly independent but with wedge product 0. Then there is a non-singular algebraic curve C, a regular morphism

φ: X → C,

and differentials of the first kind ω′_i, such that the pullbacks

φ*(ω′_i) = ω_i.

This result is due to Guido Castelnuovo and Michele de Franchis (1875–1946). (The converse, that two such pullbacks would have wedge 0, is immediate.)

See also: de Franchis theorem