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Abstract

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