The moduli space of totally marked degree two rational maps

Anupam Bhatnagar¹

¹ Department of Mathematics Borough of Manhattan Community College The City University of New York 199 Chambers Street New York, NY 10007, U.S.A.

Acta Arithmetica, Tome 167 (2015) no. 3, pp. 251-260

Résumé

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

Détail

DOI : 10.4064/aa167-3-3

Keywords: rational map phi mathbb mathbb along ordered list fixed critical points called totally marked rational map space rat totally marked degree rational maps parametrized affine subset mathbb consider natural action rat induced action mathbb prove quotient space rat exists scheme quotient isomorphic del pezzo surface isomorphism being defined mathbb