The moduli space of totally marked degree two rational maps

Volume 167 / 2015

Anupam Bhatnagar Acta Arithmetica 167 (2015), 251-260 MSC: Primary 14L30; Secondary 14L24, 14D22, 37P45. DOI: 10.4064/aa167-3-3


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


  • Anupam BhatnagarDepartment of Mathematics
    Borough of Manhattan Community College
    The City University of New York
    199 Chambers Street
    New York, NY 10007, U.S.A.

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