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Could anybody help me with the following question ?

Assume we are given:

(1) a finite order (linear) automorphism $g$ of the projective space $\mathbb{P}^r$,

(2) a closed algebraic subvariety $Z \subset \mathbb{P}^r$ of codimension at least 2.

Is it always possible to find a line in $\mathbb{P}^r$ which is stable under $g$ and does not meet $Z$?

Thanks in advance!

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What do you guess? – Giovanni De Gaetano Apr 21 '12 at 14:51
More than a guess I hope this to be true – stable Apr 21 '12 at 14:56
up vote 1 down vote accepted

It is false in general. Consider the automorphism $g$ of $\mathbb{P}^2(\mathbb{C})$ defined by $g \colon [x_0,x_1,x_2] \mapsto [x_0, - x_1, i x_2]$ and take $Z \subseteq \mathbb{P}^2(\mathbb{C})$ made up of $[1,0,0], [0,1,0], [0,0,1]$, i.e. the fixed points of $g$. But $\{x_0 = 0\}, \{x_1 = 0 \}, \{x_2 = 0 \}$ are the only lines which are $g$-invariant.

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Thanks for the example. In that case, one can embed $\mathbb{P}^2(\mathbb{C})$ in $\mathbb{P}^5(\mathbb{C})$ by the Veronese map of degree two. Then $g$ extends to a finite order automorphism of $\mathbb{P}^5(\mathbb{C})$ and one can find an invariant line there that does not meet the image of $Z$. Is that always possible for a Veronese embedding of sufficiently large degree? – stable Apr 21 '12 at 20:47

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