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 (1875–1946).
The converse, that two such pullbacks would have wedge 0, is immediate.
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 (1875–1946).
The converse, that two such pullbacks would have wedge 0, is immediate.