The Moduli Space of Totally Marked Degree Two Rational Maps
Anupam Bhatnagar

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 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, can be parametrized by an affine open subset of . We consider the natural action of on induced from the action of on and prove that the quotient space exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over .
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