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.

This algebraic geometry–related article is a stub.