Castelnuovo–de Franchis theorem

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 ω1 and ω2 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 φ: X → C, and differentials of the first kind ω′1 and ω′2 on C such that φ*(ω′1) = ω1 and φ*(ω′2) = ω2.

This result is due to Guido Castelnuovo and Michele de Franchis (1875–1946).

Il contrario, that two such pullbacks would have wedge 0, è immediato.

See also de Franchis theorem References Coen, S. (1991), Geometry and Complex Variables, Lecture Notes in Pure and Applied Mathematics, vol. 132, CRC Press, p. 68, ISBN 9780824784454. Categorie: Algebraic surfacesTheorems in geometry