A regularization theorem for bounded-degree self-maps

Let K be an algebraically closed field of arbitrary characteristic and let X be an irreducible projective variety over K. Let G(X) be a bounded-degree subgroup. We prove that there exists an irreducible projective variety Y birational to X, such that every element of G becomes an automorphism of Y after the birational transformation. If K=C, this result is stated in [Can14, Theorem 2.5] and the proof backs to [HZ96, Section 5]. The proof in [HZ96] is not purely algebraic. Inheriting the methods in [HZ96], we give a purely algebraic proof of this statement in arbitrary characteristic. We will also discuss a corollary of this result which is useful in arithmetic dynamics.