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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).

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).

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 integer 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).

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).

The group of symplectomorphisms $Aut(X)$ of a K3 is the group $O(\Lambda)$ of automorphisms of its period lattice $\Lambda=H^{1,1}(M,\Z)$. For each integer cohomology class $\eta\in H^{1,1}(M,\Z)$, 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 of classes $\eta$ with $q(\eta, \eta)$ with finitely many orbits).

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 integer 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).