TITLE: Action of automorphisms of a $K3$ surface on its $(-2)$-curves
QUESTION [5 upvotes]: Consider a complex $K3$ surface $X$ and take its group of automorphisms $Aut(X)$. It is a known fact that the action of $Aut(X)$ on the set of rational $-2$ curves of $X$ has only finite number of orbits.
Questions.  What kind of ideas one has to use to prove this fact? Is there some nice exposition? Does this fact follows somehow from global Torelli theorem for $K3$'s?

REPLY [13 votes]: The group of symplectomorphisms $Aut(X)$ of a K3 is the group $O(\Lambda)$ of 
automorphisms of its period lattice $\Lambda=H^{1,1}(M,{\Bbb Z})$. For each 
(-2)-cohomology class $\eta\in H^{1,1}(M,{\Bbb Z})$, either $\eta$ or $-\eta$ is
represented by a curve (this follows from the Riemann-Roch formula). This curve
is unique, because its self-intersection is negative. Therefore, finiteness of the
number of orbits of $Aut(X)$ on curves is equivalent to the finiteness of the number of orbits of $O(\Lambda)$ on the set of (-2)-classes in $\Lambda$. This result follows from an answer to a question that I asked on Mathoverflow:
orbits of automorphism group for indefinite lattices
(automorphism group $O(\lambda)$ of any lattice $(\Lambda, q)$acts on the set $R_C$ of classes 
$\eta$ with $q(\eta, \eta)=C$ with finitely many orbits).