The Moduli Space of Totally Marked Degree Two Rational Maps

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

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