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 ...
, Noether's theorem on rationality for surfaces is a classical result of
Max Noether 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, giving a criterion for a
rational surface. Let ''S'' be an algebraic surface that is
non-singular and projective. Suppose there is a morphism φ from ''S'' to the
projective line, with ''general fibre'' also a projective line. Then the theorem states that ''S'' is rational.
See also
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Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .
Definition
The Hirzebruch surface \Sigma_n is the \mathbb^1-bundle, called a Projective bundle, over \mathbb^1 associated to the sheaf\mathca ...
*
List of complex and algebraic surfaces
References
Castelnuovo’s Theorem
Notes
Algebraic surfaces
Theorems in algebraic geometry
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