Automorphisms of Smooth Hypersurfaces with Fixed Loci of Codimension at Most Two

TL;DR

This paper investigates automorphisms of smooth hypersurfaces with fixed loci of codimension at most two, revealing restrictions on automorphism orders and exploring the rationality of associated quotient spaces.

Contribution

It establishes new restrictions on automorphism orders based on fixed locus codimension and studies the rationality of quotient spaces for certain automorphisms.

Findings

Fixed locus codimension limits automorphism orders

Automorphisms with orders multiple of d-1 or d relate to quotient rationality

Classification of automorphism orders is refined under fixed locus constraints

Abstract

We study automorphisms of smooth hypersurfaces in projective space $\mathbb{P}^{n+1}$ whose fixed loci have codimension at most two for $n\geq2$. While classifications of possible orders of automorphisms are known, our aim is to explore the relationship between the order of an automorphism and its algebraic and geometric properties. In this paper, we show that the assumption on the fixed locus restricts the possible orders of automorphisms. Moreover, when the fixed locus has codimension at most two, we investigate the rationality of quotient spaces associated with automorphisms whose orders are multiples of $d-1$ or $d$, where $d$ denotes the degree of the hypersurface.