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<p>We prove that over an algebraically closed field of arbitrary characteristic, a bounded-degree group of birational self-maps of an irreducible projective variety can be simultaneously regularized. In other words, there exists another projective variety that is birational to the original one such that every element of the bounded-degree group turns into an automorphism after the birational transformation. Over the field of complex numbers, this result was stated in an article by Cantat (2014) and the proof originates from a work of Huckleberry and Zaitsev (1996). The original proof, however, is not purely algebraic. By adapting their methods, we provide a purely algebraic proof of this statement in arbitrary characteristic. Moreover, we discuss a corollary of this result that is useful in arithmetic dynamics.</p>

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A regularization theorem for bounded-degree self-maps

  • 期刊论文

  • 发表日期：

    2025年4月29日

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  • She Yang

摘要

We prove that over an algebraically closed field of arbitrary characteristic, a bounded-degree group of birational self-maps of an irreducible projective variety can be simultaneously regularized. In other words, there exists another projective variety that is birational to the original one such that every element of the bounded-degree group turns into an automorphism after the birational transformation. Over the field of complex numbers, this result was stated in an article by Cantat (2014) and the proof originates from a work of Huckleberry and Zaitsev (1996). The original proof, however, is not purely algebraic. By adapting their methods, we provide a purely algebraic proof of this statement in arbitrary characteristic. Moreover, we discuss a corollary of this result that is useful in arithmetic dynamics.