Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point of Small Order

The surface corresponding to the moduli space of quadratic endomorphisms of P-1 with a marked periodic point of order n is studied. It is shown that the surface is rational over Q when n <= 5 and is of general type for n= 6. An explicit description of the n = 6 surface lets us find several infinite families of quadratic endomorphisms f : P-1 -> P-1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n > 3.