# 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.[1]

## References

## Notes

- Kurke, G. (1972). "The castelnuovo criterion of rationality" (PDF).
*Mathematical Notes of the Academy of Sciences of the USSR*.**11**: 20–23. doi:10.1007/BF01366911.

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