Interest in algebraic surfaces with pg = h0(O(K)) = 0 goes back to the work of Enriques and Castelnuovo in the 19th Century. After Clebsch had proved that curves with pg = 0 are rational, these authors considered the analogous question for surfaces. It was clear to them that in this case the irregularity q = h1(O(K)) has to be controlled as well.

In 1894 Enriques constructed his now famous surface, which is irrational with q= pg = 0? disproving the most obvious rationality criterion. Two years later Castelnuovo proved that the modified conditions q = P2 = p2 = 0 do imply rationality. Thus he substituted the second plurigenus from pk = h0(O(Kk)) for the first. (For the Enriques surface K is a 2-torsion bundle, so the bigenus is one.)

Over the next forty years more examples of irrational surfaces with q = pg = 0 were constructed. Like the Enriques surface they were all elliptic. Only in 1931 did Godeaux [G] find a surface of general type with these invariants. His construction was disarmingly simple: divide the Fermat quintic in CP3 by the standard free Z5- action on the coordinates. Campedelli also gave an example of a surface of general type, introducing his “double plane” construction. This has H1(X, Z) = Z32.