The Moduli Space of Totally Marked Degree Two Rational Maps

Abstract: A rational map $\phi: \mathbb{P}¹ \to \mathbb{P}<sup>1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat<sup>{tm}_2$ can be parametrized by an affine open subset of $(\mathbb{P}<sup>1)<sup>5$. We consider the natural action of $SL_2$ on $Rat<sup>{tm}_2$ induced from the action of $SL_2$ on $(\mathbb{P}<sup>1)<sup>5$ and prove that the quotient space $Rat<sup>{tm}_2/SL_2$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over $\mathbb{Z}[1/2]$.