Noether's theorem on rationality for surfaces
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. See also
Hirzebruch surface
List of complex and algebraic surfaces

References
Castelnuovo’s Theorem

Notes

Algebraic surfaces
Theorems in algebraic geometry