We show that the degrees of rational endomorphisms of very general complex Fano and Calabi–Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree \(d\ge \lceil 5(n+3)/6\rceil \) are not birational to Jacobian fibrations of dimension one. A key part of the argument is to resolve singularities of general \(\mu _{p}\)-covers in mixed characteristic p.

中文翻译：

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Fano 超曲面的有理自同态

我们证明了非常一般的复杂 Fano 和 Calabi-Yau 超曲面的有理自同态度通过特化特征 p 满足某些同余条件。作为推论，我们表明非常一般的n维超曲面\(d\ge \lceil 5(n+3)/6\rceil \)与一维雅可比纤维不是双有理的。论证的关键部分是解决混合特征 p 中一般\(\mu _{p}\)覆盖的奇点。