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I am looking for some references for the following statement:

Let $G$ be a linearly reductive algebraic group acting on a quasi-projective scheme $X$, over an algebraically closed field $K$. Let $L$ be an ample line bundle on $X$. Then there exists a sufficiently large $l>>0$ such that $L^{\otimes l}$ is $G$-linearized.

This one was suggested to me by a friend of mine, but he doesn't know any references. I tried to search it on some classic books, as GIT or Newstead (Introduction to moduli and orbit spaces), but without any results. If you know another similar statement in a book or a reference where I can find also a general treatment, I would be very happy.

Thank you!

P.S.: I trust that this is true, but of course I am not so sure. I am looking for a reference also to see if the hypothesis are right or no.

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  • $\begingroup$ You forgot an obvious necessary condition, namely that some power $L^m$ of $L$ should be $G$-invariant (i.e. $g^*L^m\cong L^m$ for all $g$ in $G$). That this is equivalent to your statement is Proposition 1.5 in Mumford's GIT. $\endgroup$
    – abx
    Commented Aug 9, 2014 at 14:29
  • $\begingroup$ abx's comment immediately provides counterexamples when $G$ is not connected. There are also counterexamples when $G$ is connected and $X$ not normal, but none if $G$ is connected and $X$ is normal : see question mathoverflow.net/questions/109310/… , its answer and the references therein. $\endgroup$
    – Olivier Benoist
    Commented Aug 9, 2014 at 14:34
  • $\begingroup$ It seems to me that these comments could be left as answers $\endgroup$
    – Yemon Choi
    Commented Aug 9, 2014 at 22:46

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