  While the Chow groups of 0-dimensional cycles on the moduli spaces of
Deligne-Mumford stable pointed curves can be very complicated, the span of the
0-dimensional tautological cycles is always of rank 1. The question of whether
a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is
subtle. Our main results address the question for curves on rational and K3
surfaces. If C is a nonsingular curve on a nonsingular rational surface of
positive degree with respect to the anticanonical class, we prove
[C,p_1,...,p_n] is tautological if the number of markings does not exceed the
virtual dimension in Gromov-Witten theory of the moduli space of stable maps.
If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is
tautological if the number of markings does not exceed the genus of C and every
marking is a Beauville-Voisin point. The latter result provides a connection
between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1
tautological 0-cycles on K3 surfaces. Several further results related to
tautological 0-cycles on the moduli spaces of curves are proven. Many open
questions concerning the moduli points of curves on other surfaces (Abelian,
Enriques, general type) are discussed.
