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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

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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 Starr Aug 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) –  user12770 Aug 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. –  user12770 Aug 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? –  user12770 Aug 4 '12 at 6:25

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