Some remarks on finiteness of extremal rays of divisorial type

Abstract

Let $X$ be a normal $\mathbf{Q}$-factorial projective variety with at most log canonical singularities. We shall give a sufficient condition for the existence of at most finitely many $K_{X}$-negative extremal rays $R(\subset \overline{\mathrm{NE}}(X))$ of divisorial type. As an application, we show that for a nonisomorphic surjective endomorphism $f\colon X\to X$ of a normal projective $\mathbf{Q}$-factorial terminal 3-fold $X$ with $\kappa(X) > 0$, a suitable power $f^{k}\ (k > 0)$ of $f$ descends to a nonisomorphic surjective endomorphism $g\colon X_{\textit{min}}\to X_{\textit{min}}$ of a minimal model $X_{\textit{min}}$ of $X$.