The Moduli Space of Totally Marked Degree Two Rational Maps

Abstract

A rational map $\varphi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ 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, $Ra{t}_2^{tm}$ can be parametrized by an affine open subset of $({\mathbb{P}}^1{)}^5$. We consider the natural action of $S{L}_2$ on $Ra{t}_2^{tm}$ induced from the action of $S{L}_2$ on $({\mathbb{P}}^1{)}^5$ and prove that the quotient space $Ra{t}_2^{tm}/S{L}_2$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over $\mathbb{Z}[1/2]$.

11 pages, To Appear in Acta Arithmetica