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

Taro Hayashi, Ryoichi Suzuki

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.

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

Abstract

We study automorphisms of smooth hypersurfaces in projective space whose fixed loci have codimension at most two for . 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 or , where denotes the degree of the hypersurface.
Paper Structure (11 sections, 28 theorems, 166 equations)

This paper contains 11 sections, 28 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorems \ref{['1.1']} and \ref{['1.2']}
  4. Type I
  5. Types II and III
  6. Type II
  7. Type III
  8. Types IV, V, and VI
  9. Type IV
  10. Type V
  11. Type VI

Key Result

Theorem 1.1

Let $\mathcal{X} \subset \mathbb P^{n+1}$ be a smooth hypersurface of degree $d$ for $n\geq2$, and let $g\in \mathrm{PGL}(n+2,k)$ be an automorphism of $\mathcal{X}$. We assume that the fixed locus $\mathrm{Fix}(g)$ has codimension one in $\mathcal{X}$. Then the order of $g$ divides Moreover, if the order of $g$ divides $d-2$, then $n=2$. We assume that the fixed locus $\mathrm{Fix}(g)$ has codim

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • ...and 37 more