Moduli spaces of quadratic rational maps with a marked periodic point of small order

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The surface corresponding to the moduli space of quadratic endomorphisms of P¹ 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: P¹ → P¹ 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.