I would like to know for which projective smooth surfaces over a finite field there exists a dominant rational map from a product of curves to the surface

That is a bit open-ended. Are you looking for necessary conditions? Are you looking for sufficient conditions? Are you looking at a specific class of surfaces, e.g., hypersurface in $\mathbb{P}^3$? Is there a guess about this problem that you are trying to test?
– Jason StarrAug 2 '12 at 19:16

Sufficient/Necessary conditions, counter-examples, references, all will be welcome since I don't know how to attack the question. (I hope that most projective smooth surfaces over a finite field satisfy this condition)
– user12770Aug 3 '12 at 3:35

Well my questions seems to not provoke a lot of enthusiasm! I will be happy enough if somebody give me an example of projective smooth surface over a finite field not dominated by a product of curves.
– user12770Aug 3 '12 at 15:04

non DPC surfaces have been found by CHAD SCHOEN, Int. J. Math., 07, 541 (1996), VARIETIES DOMINATED BY PRODUCT VARIETIES Is there a way to characterize non DPC projective smooth surfaces over finite fields?
– user12770Aug 4 '12 at 6:25