On the Constant Reductions of Automorphism Groups of valued Function Fields

TL;DR

This paper explores how automorphism groups of one-variable function fields behave under constant reductions, generalizing a classical theorem by Deligne and Mumford within Deuring's framework.

Contribution

It extends the understanding of automorphism group reductions in function fields, providing a broader generalization of existing theorems in algebraic geometry.

Findings

Established a generalized injective homomorphism for automorphism groups under constant reductions.

Connected classical results with Deuring's theory to broaden applicability.

Enhanced understanding of automorphism group behavior in valued function fields.

Abstract

In this paper, we investigate properties of automorphism groups of function fields in one variable in relation to its reductions with respect to special valuations. In 1969, Deligne and Mumford proved that there exists a natural injective homomorphism between the automorphism groups of $\mathcal{X}_{\eta}$ and any special fibre of $\mathcal{X}.$ Here, we give a generalisation of this theorem in function field setting of Deuring's theory of constant reductions.