The Moduli Space of Totally Marked Degree Two Rational Maps

TL;DR

This paper studies the moduli space of degree two rational maps with marked points, showing it forms a scheme and is isomorphic to a Del Pezzo surface over certain integers, revealing geometric structure and symmetries.

Contribution

It establishes the existence of the quotient space of totally marked degree two rational maps as a scheme and identifies it with a Del Pezzo surface over , providing a geometric classification.

Findings

The quotient space exists as a scheme.

It is isomorphic to a Del Pezzo surface.

The isomorphism is defined over .

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 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 $\mathbb{Z}[1/2]$.