Noether's theorem on rationality for surfaces

{{Short description|Theorem}}

In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, 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.{{cite journal | doi=10.1007/BF01366911 | volume=11 | title=The castelnuovo criterion of rationality | year=1972 | journal=Mathematical Notes of the Academy of Sciences of the USSR | pages=20–23 | last1 = Kurke | first1 = G.}}