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Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map then we define $Aut(\phi)=\{f \in PGL_2(\Bbb C): f^{-1}\phi f(z)=\phi(z)\}$

Here, in general, the definition of a rational map is: Let $\mathbb{P}^n$ and $\mathbb{P}^m$ be projective spaces. If $m$ homogeneous polynomials in $n+1$ variables of the same degree "d" give a partially defined map from $\mathbb{P}^n$ to $\mathbb{P}^m$ then this map is called a rational map. We can exactly define this on a projective variety as well.

Now for $\mathbb{P}^1$ we can prove that:

Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map of degree $d \geq 2$. Then $Aut(\phi)$ is a finite subgroup of $ PGL_2(\Bbb C)$ and its order is bounded by a function of $d$.

This proof can be done by using the fact that $f\in PGL_2(\Bbb C)$ is basically a Mobius transformation and once it fixes three points it is constant and once we know this result we can manipulate the periodic points of $\phi$ to have an upper bound. If you ask for details I can fill in the gaps.

The problem arises if we go to higher dimension say $\mathbb{P}^2$ or $\mathbb{P}^n, n\geq 2$. Then we will have at least 3 homogeneous polynomials in at least $3$ variables of the same degree. Can we extend this result and have some bounds? Any hint or result would be much appreciated. Even if you give some sequential hints I can try to fill the gaps and reach out for help in the comments.

Baby Elephant
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  • I suspect if you only consider endomorphisms things work the same: endomorphisms only have finitely many fixed points ($d^n$ counted with multiplicity), and any automorphism $f$ commuting with $\phi$ must map them to each other. But an element of $PGL_n$ fixing $n+1$ points in the identity – Albert Sep 25 '19 at 09:24

1 Answers1

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For $f:\mathbb P^n\to\mathbb P^n$ of degree $d$, there is a bound for $\operatorname{Aut}(f)$ that depends only on $n$ and $d$. This is proven in:

  • Levy, Alon, The space of morphisms on projective space, Acta Arith. 146 (2011), 13-31

Additional information of interest may be found in

  • de Faria, Joao Alberto and Hutz, Benjamin, Automorphism groups and invariant theory on $\mathbb{P}^N$, J. Algebra Appl. 17 (2018).
  • Cai, Julia and Hutz, Benjamin and Mayer, Leo and Weinreich, Max, Automorphism Groups of Endomorphisms of $\mathbb{P}^1(\overline{\mathbb{F}}_p)$, 2020, arXiv:2003.12113
Joe Silverman
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