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I am interested in the following question: let $n\geq 2$ be an integer and consider a product of $n$ copies of the projective line:

$X=\mathbb{P}^1 \times \cdots \times \mathbb{P}^1$

(say everything is defined over an algebraically closed field of characteristic zero). Consider the automorphism $\sigma: X \to X$ sending $(t_1, \ldots, t_n)$ to $(t_n, t_1, \ldots, t_{n-1})$. Is it possible to construct equivariant morphism from $X$ to $\mathbb{P}^1$ for the trivial action on the later. If so, which are the easier ones? I guess the answer has to do with symmetric functions but I have not managed to get things written down.

Thanks for your help!

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The quotient of $\mathbb P^1\times \mathbb P^1$ by the symmetric group $S_2$ is isomorphic (because of symmmetric functions, as you guessed) to $\mathbb P^2$.
So any equivariant morphism $\mathbb P^1\times \mathbb P^1\to \mathbb P^1$ factorizes through $\mathbb P^2 \to \mathbb P^1$.
But these latter morphisms are well known to be constant.

Any morphism $f:\mathbb P^1\times \mathbb P^1\to \mathbb P^1 $ such that for all $x,y\in \mathbb P^1$ we have $f(x,y)=f(y,x)$ is constant.

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