In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Castelnuovo–de Franchis theorem is a classical result on complex
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s. Let ''X'' be such a surface, projective and
non-singular, and let
:ω
1 and ω
2
be two
differentials of the first kind
In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
on ''X'' which are linearly independent but with
wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
0. Then this data can be represented as a
pullback of an
algebraic curve: there is a non-singular algebraic curve ''C'', a
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
:φ: ''X'' → ''C'',
and differentials of the first kind ω
1 and ω
2 on ''C'' such that
:φ*('
1) = ''ω''
1 and φ*('
2) = ''ω''
2.
This result is due to
Guido Castelnuovo
Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also signi ...
and
Michele de Franchis (1875–1946).
The converse, that two such pullbacks would have wedge 0, is immediate.
See also
*
de Franchis theorem
References
*.
*
{{DEFAULTSORT:Castelnuovo-De Franchis Theorem
Algebraic surfaces
Theorems in geometry