Finite Morphisms between Projective Varieties and Skeleta

TL;DR

This paper investigates finite morphisms between irreducible projective varieties over non-archimedean fields by analyzing induced morphisms on their analytifications, revealing structural properties of fibers and associated neighborhoods.

Contribution

It introduces a canonical association of points in analytifications and establishes a deformation retraction with fiberwise size invariance for neighborhoods related to finite morphisms.

Findings

Existence of a deformation retraction onto a finite simplicial complex.

Constant size of neighborhoods along fibers of the retraction.

Structural insights into fibers of finite morphisms in non-archimedean geometry.

Abstract

In this paper we study finite morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let $V'$ and $V$ be irreducible, projective varieties over an algebraically closed, non- archimedean valued field $k$ and $\phi$ be a finite morphism $\phi : V' \to V$. Let $x \in V^{an}(L)$, where $L/k$ is an algebraically closed complete non-archimedean valued field extension. We associate canonically to $x$ an $L$-point of the space $(V \times_k L)^{an}$ which lies on the fiber over $x$ and denote this point $x_L$. The embedding of $V$ into some $n$-dimensional projective space defines in a natural way a family of open neighbourhoods $\mathcal{O}_{x_L}$ in $(V \times_k L)^{an}$ of $x_L$. Each element of this family is parametrized by an $(n + 1)^2$-tuple which quantifies its size. Of particular interest to us will be those elements $O$ of the set $\mathcal{O}_{x_L}$ whose preimage for the morphism $(\phi \times id_L)^{an}$ decomposes into the disjoint union of homeomorphic copies of $O$ via $(\phi \times id_L)^{an}$. Let $\mathcal{G}_{x_L} \subset \mathcal{O}_{x_L}$ denote the sub collection of elements of this form. Theorem 1.3 shows that there exists a deformation retraction of the space $V$ onto a finite simplicial complex such that along the fibers of the retraction the size of the largest element belonging to $\mathcal{G}_{x_L}$ is constant.