Castelnuovo-de Franchis theorem
- Castelnuovo-de Franchis theorem
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
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