Complex sentence: The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo's theorem implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both the arithmetic genus and the second plurigenus.
Split into simpler sentences: The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo's theorem implies that for complex surfaces unirational implies rational. For the field of complex numbers, Guido Castelnuovo had characterised the unirational and the rational varieties amongst algebraic surfaces by the same criterion: the vanishing of both the arithmetic genus and the second plurigenus.