Let K denote a number field, S a finite set of places of K, and ϕ: ℙⁿ → ℙⁿ a rational morphism defined over K. The main result of this paper states that there are only finitely many twists of ϕ defined over K which have good reduction at all places outside S. This answers a question of Silverman in the affirmative.

A rational map ϕ: ℙ¹ → ℙ¹ along with an ordered list of fixed and critical points is called a totally marked rational map. The space $Rat{₂}^{t}m$ of totally marked degree two rational maps can be parametrized by an affine open subset of (ℙ¹)⁵. We consider the natural action of SL₂ on $Rat{₂}^{t}m$ induced from the action of SL₂ on (ℙ¹)⁵ and prove that the quotient space $Rat{₂}^{t}m/SL₂$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over ℤ[1/2].