A6-invariant curves of genera 10 and 19.

We study smooth curves on which the alternating group A 6 acts faithfully. Let V ⊂ PGL (3 , C) be the Valentiner group, which is isomorphic to A 6 . We see that there are integral V -invariant curves of degree 12 which have geometric genera 10 and 19. On the other hand, if A 6 acts faithfully on a curve C of genus 10 or 19, then we give an explicit description of the extension k (C / A 5) / k (C / A 6) for any icosahedral subgroup A 5 . Using this, we show the uniqueness of smooth projective curves of genera 10 and 19 whose automorphism groups contain A 6 . [ABSTRACT FROM AUTHOR]