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A minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field

J.-P. Serre

Collège de France, Paris Cedex

Abstract: Let $\mathrm{Cr}(k)=\operatorname{Aut}k(X,Y)$ be the Cremona group of rank 2 over a field $k$. We give a sharp multiplicative bound $M(k)$ for the orders of the finite subgroups $A$ of $\mathrm{Cr}(k)$ such that $|A|$ is prime to $\mathrm{char}(k)$. For instance $M(\mathbf Q)=120960$, $M(\mathbf F_2)=945$ and $M(\mathbf F_7)=847065600$.

Key words and phrases: Cremona group, algebraic torus, Del Pezzo surface, conic bundle.

MSC: Primary 14E07; Secondary 14J26

Received: May 14, 2008

Language: English

DOI: 10.17323/1609-4514-2009-9-1-183-198

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