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arXiv · 2211.16380

Boundedness of finite morphisms onto Fano manifolds with large Fano index

Feng Shao·Guolei Zhong
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Abstract

Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such that the Fano index of $X$ is greater than 1. On the one hand, when both $X$ and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is bounded in terms of $X$ and $Y$ unless $X\cong\mathbb{P}^4$; hence, such $X$ does not admit any non-isomorphic surjective endomorphism. On the other hand, when $X=Y$ is either a fourfold or a del Pezzo manifold, we prove that, if $f$ is an int-amplified endomorphism, then $X$ is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms.

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Feng Shao, Guolei Zhong. 2023-11-27. Boundedness of finite morphisms onto Fano manifolds with large Fano index. https://doi.org/10.1016/j.jalgebra.2023.10.030

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