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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 complex projective space $\mathbb{P}^r$,

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

The problem is to find lines in $\mathbb{P}^r$ which are stable under $g$ and do not meet $Z$. This is not always possible. However, one can embed $\mathbb{P}^r$ in a bigger dimension projective space $\mathbb{P}^N$ by the Veronese embedding of degree $d$ and extend $g$ to a finite order automorphism of $\mathbb{P}^N$. Is it true, for sufficiently large $d$, that there exists a $g$-invariant line in $\mathbb{P}^N$ which does not meet the image of $Z$?

Thanks in advance!

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  • $\begingroup$ Yes, this is true. Since the order of $g$ is fixed, if $N$ is large there will be linear subspaces $L$ of large dimension preserved pointwise by $d$. Consider any line in such a subspace not meeting the image of $Z$. $\endgroup$
    – naf
    Commented Apr 22, 2012 at 7:19
  • $\begingroup$ Thanks ulrich! Could you please give more details? $\endgroup$
    – reference
    Commented Apr 22, 2012 at 7:26
  • $\begingroup$ Since $g$ is of finite order, you can assume it is given by a diagonal matrix in $SL_{r+1}$ whose entries are roots of unity. It is clear from the definition of the Veronese that $g$ will also act diagonally on $\mathbb{P}^N$, the space corresponding to the Veronese embedding. The diagonal entries are again roots of unity of the same (or smaller) order. Since there are only finitely many such, if $N$ is large there will be many repeated entries. The subspace corresponding to the repeated entries of any root of unity is fixed pointwise. $\endgroup$
    – naf
    Commented Apr 23, 2012 at 13:03

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