Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smooth Fano 3-folds satisfying Condition (A)

Hamid Abban, Ivan Cheltsov, Takashi Kishimoto, Frederic Mangolte

TL;DR

This work classifies which smooth Fano $3$-folds satisfy Condition $(A)$, where every finite abelian subgroup of the automorphism group fixes a point, across all 105 deformation families. The authors combine prior results on many families with a meticulous case-by-case analysis, using equivariant birational geometry and explicit automorphism actions to determine fixed-point behavior, while also producing explicit counterexamples in other families. They derive arithmetic consequences, including statements about unirationality over subfields and the existence of rational points, and connect geometric fixed-point properties to arithmetic obstructions via Lang–Nishimura-type arguments. The Appendix further clarifies the degree-$14$ case, showing the equivalence of several natural conditions for $k$-points and unirationality, and providing explicit constructions to illustrate the landscape of $k$-points and unirationality across families.

Abstract

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).

Smooth Fano 3-folds satisfying Condition (A)

TL;DR

This work classifies which smooth Fano -folds satisfy Condition , where every finite abelian subgroup of the automorphism group fixes a point, across all 105 deformation families. The authors combine prior results on many families with a meticulous case-by-case analysis, using equivariant birational geometry and explicit automorphism actions to determine fixed-point behavior, while also producing explicit counterexamples in other families. They derive arithmetic consequences, including statements about unirationality over subfields and the existence of rational points, and connect geometric fixed-point properties to arithmetic obstructions via Lang–Nishimura-type arguments. The Appendix further clarifies the degree- case, showing the equivalence of several natural conditions for -points and unirationality, and providing explicit constructions to illustrate the landscape of -points and unirationality across families.

Abstract

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).
Paper Structure (4 sections, 9 theorems, 63 equations)

This paper contains 4 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Proof of Proposition B
  4. Unirationality of smooth Fano 3-folds of degree $14$

Key Result

Lemma 2.1

Let $A$ be a finite abelian subgroup of the group $\mathrm{Aut}(X)$. Suppose that $X$ is contained in one of the following families: № 1.11, № 2.1, № 2.15, № 3.15, № 5.1. Then $A$ fixes a point in $X$.

Theorems & Definitions (57)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6: Gushel-Mukai 3-folds
  • Example 2.7: Beauville
  • Example 2.8
  • Example 2.9
  • ...and 47 more