Abstract
The aim is to give a simple proof of this famous and important theorem which is fundamental in the Enriques-Kodaira classification of complex surfaces. Modern textbooks like Griffiths and Harris(Principles of Algebraic Geometry) or Arnaud Beauville(Complex Algebraic Surfaces) usually present the proof by Kodaira, in the fourth of his series of papers on classification of Complex Algebraic Surfaces[1]. Here we present a proof more in the spirit of Mori theory. I do not claim any originality, to experts this proof must be well known, though I have not seen it in books. As a consequence we get the version of Lüroth theorem for two dimensions. The higher dimensional version of the Lüroth theorem is false by the counterexamples of Griffiths-Clemens(using the intermediate Jacobian), Michael Artin-Mumford(using ideas of the Indian Algebraic Geometer C. P. Ramanujam who died tragically at a young age) and Ishkovskih and Manin who independently found examples around 1971. We also comment briefly on Enriques surfaces which provides an example to show that one cannot relax the conditions in Castelnuovo's theorem.