Noether's Theorem On Rationality For Surfaces
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Noether's theorem on rationality for surfaces is a classical result of
Max Noether Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the ...
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 In algebraic geometry, a branch of mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sc ...
. Let ''S'' be an algebraic surface that is
non-singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular ...
and projective. Suppose there is a morphism φ from ''S'' to the
projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...
, with ''general fibre'' also a projective line. Then the theorem states that ''S'' is rational.


See also

* Hirzebruch surface *
List of complex and algebraic surfaces This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification. Kodaira dimension −∞ Rational surfaces * Projective plane Qu ...


References


Castelnuovo’s Theorem


Notes

Algebraic surfaces Theorems in algebraic geometry {{algebraic-geometry-stub