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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).

The converse, that two such pullbacks would have wedge 0, is immediate.
See also

de Franchis theorem

References

Coen, S. (1991), Geometry and Complex Variables, Lecture Notes in Pure and Applied Mathematics 132, CRC Press, p. 68, ISBN 9780824784454.
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Mathematics Encyclopedia

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