Castelnuovo–de Franchis theorem
In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let
be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pullback of an algebraic curve: there is a non-singular algebraic curve C, a morphism
and differentials of the first kind ω′1 and ω′2 on C such that
This result is due to Guido Castelnuovo and Michele de Franchis.
The converse, that two such pullbacks would have wedge 0, is immediate.
