Termination of extremal rays of divisorial type for the power of étale endomorphisms (Q2010763)

The paper under review considers the following problem: let \(f:X\to X\) be a non-trivial étale endomorphism such that there exists a \(K_X\)-negative extremal ray of fiber type, then whether any \(K_X\)-negative extremal ray of divisorial type is preserved by some \((f^k)_*: N_1(X)\to N_1(X)\). The main result gives an affirmative answer to this question when \(\rho(X)=2\). When the Picard number is higher, there is a counterexample given by blowing up 9 points on the projective plane.