minwaardekn

2022-06-25

Suppose $f:{\mathbb{P}}^{2}\to {\mathbb{P}}^{2}$ is rational such that $f\circ f=\mathbb{I}\mathbb{d}$. Then is it true that $f$ must be linear?

It feels true due to the degree which increases, but some things might cancel out.

Suppose we have a smooth curve $C$ of genus $g\ge 1$ with a rational function $g:C\to C$ with the same property. Does it always rise to an $f$ on ${\mathbb{P}}^{2}$ whose restriction is $g$? Does it imply that $g$ has to be linear too? This on the other side seems wrong to me.

It feels true due to the degree which increases, but some things might cancel out.

Suppose we have a smooth curve $C$ of genus $g\ge 1$ with a rational function $g:C\to C$ with the same property. Does it always rise to an $f$ on ${\mathbb{P}}^{2}$ whose restriction is $g$? Does it imply that $g$ has to be linear too? This on the other side seems wrong to me.

jmibanezla

Beginner2022-06-26Added 17 answers

Here is an involution

$(x,y,z)\mapsto ({x}^{2}-yz,{y}^{2}-xz,{z}^{2}-xy)$

It is the inversion wr to the conic $xy+xz+xz=0$ with inversion center (1,1,1)

Projective rational involutions appear in other instances, think of the map $A\mapsto {A}^{-1}$ for matrices, which in projective coordinates can be given as $A\mapsto \mathrm{adj}A$, from a matrix to its adjugate. This map, restricted to certain subalgebras of matrices, again gives an involution.

Still, I cannot find any rational involutions of ${\mathbb{P}}^{2}$ of degree $>2$. Also, I can't find rational involutions of ${\mathbb{P}}^{n}$ of degree $>n$.

$(x,y,z)\mapsto ({x}^{2}-yz,{y}^{2}-xz,{z}^{2}-xy)$

It is the inversion wr to the conic $xy+xz+xz=0$ with inversion center (1,1,1)

Projective rational involutions appear in other instances, think of the map $A\mapsto {A}^{-1}$ for matrices, which in projective coordinates can be given as $A\mapsto \mathrm{adj}A$, from a matrix to its adjugate. This map, restricted to certain subalgebras of matrices, again gives an involution.

Still, I cannot find any rational involutions of ${\mathbb{P}}^{2}$ of degree $>2$. Also, I can't find rational involutions of ${\mathbb{P}}^{n}$ of degree $>n$.

Tristian Velazquez

Beginner2022-06-27Added 7 answers

As Mohan mentioned in the comments, there exist non-linear functions whose power is $\mathbb{I}\mathbb{d}$

For example $[x:y:z]\mapsto [yz:xz:xy]$ has order two, and $[x:y:z]\mapsto [xy:yz:zx]$ has order six.

For example $[x:y:z]\mapsto [yz:xz:xy]$ has order two, and $[x:y:z]\mapsto [xy:yz:zx]$ has order six.

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