On log canonical models of the moduli space of stable pointed genus zero curves

The Deligne-Mumford moduli spaces of genus g n-pointed stable curves classify how algebraic families of Riemann surfaces may vary. Any such family corresponds to a subvariety of the associated moduli space. Because of this correspondence it is an interesting open problem to give a classification of subvarieties of our moduli spaces. There is a conjecture by Fulton regarding the structure of the cone of curves in the genus zero case. By constructing contractions to a large collection of geometric quotients parameterizing stable plane conics, I prove a special case of this conjecture. In addition, I show that the standard contractions to log-canonical models are, in many cases, equal to certain of Hassett's weighted pointed curve spaces and that much of the information about these models is encoded in our geometric quotients