A classification of degree $2$ semi-stable rational maps ${\mathbb{P}}^2\to {\mathbb{P}}^2$ with large finite dynamical automorphism group

Soit $K$ un corps algébriquement clos de charactéristique $0$. Dans cet article nous classifions les ${\mathrm{PGL}}_3\left(K\right)$-classes de conjugaison de fonctions rationelles $f:{\mathbb{P}}_K^2⤏{\mathbb{P}}_K^2$ de degré $2$ dominantes et semi-stables dont le groupe d’automorphismes

$$\mathrm{Aut}\left(f\right):=\{\phi \in {\mathrm{PGL}}_3\left(K\right):{\phi }^{-1}\circ f\circ \phi =f\}$$

est fini et d’ordre au moins $3$. En particulier, nous démontrons que $\#\mathrm{Aut}\left(f\right)\le 24$ en général, que $\#\mathrm{Aut}\left(f\right)\le 21$ pour les morphismes et que $\#\mathrm{Aut}\left(f\right)\le 6$ pour toutes excepté un nombre fini de classes de conjugaisons de $f$.

Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the ${\mathrm{PGL}}_3\left(K\right)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb{P}}_K^2⤏{\mathbb{P}}_K^2$ whose automorphism group

$$\mathrm{Aut}\left(f\right):=\{\phi \in {\mathrm{PGL}}_3\left(K\right):{\phi }^{-1}\circ f\circ \phi =f\}$$

is finite and of order at least $3$. In particular, we prove that $\#\mathrm{Aut}\left(f\right)\le 24$ in general, that $\#\mathrm{Aut}\left(f\right)\le 21$ for morphisms, and that $\#\mathrm{Aut}\left(f\right)\le 6$ for all but finitely many conjugacy classes of $f$.

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