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

We study automorphisms of smooth hypersurfaces in projective space ${\mathbb{P}}^{n+1}$ whose fixed loci have codimension at most two for $n\ge 2$. 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.