二维拟单项群作用的有理性问题

Rationality problem for quasi-monomial actions

We give a short survey of the rationality problem for quasi-monomial actions which includes Noether's problem and the rationality problem for algebraic tori, and report some results on rationality problem in three recent papers Hoshi, Kang and Kitayama [HKKi], Hoshi, Kang and Kunyavskii [HKKu] and Hoshi and Yamasaki [HY].

Rationality problem of three-dimensional monomial group actions

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in $\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of $G$ in $\mathrm{GL}(3,\mathbb{…

Rationality of Three-Dimensional Quotients by Monomial Actions

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field $K(x,y,z)^G$ is rational (= purely transcendental) over $K$. Applications of this theorem will be given.

Three-dimensional purely quasi-monomial actions

Let $G$ be a finite subgroup of $\mathrm{Aut}_k(K(x_1, \ldots, x_n))$ where $K/k$ is a finite field extension and $K(x_1,\ldots,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1, \ldots, x_n)$ is called quasi-monomial if it satisfies the following three conditions (i) $σ(K)\subset K$ for any $σ\in G$; (ii) $K^G=k$ where $K^G$ is the fixed field under the action of $G$; (iii) for any $σ\in G$ and $1 \leq j \leq n$, $σ(x_j)=c_j(σ)\prod_{i=1}^n x_i^{a_{ij}}$ where $c_j(σ)\in K^\times$ and $[a_{i,j}]_{1\le i,j \le n} \in GL_n(\mathbb{Z})$. A quasi-monomi…

Rationality properties of manifolds containing quasi-lines

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X' that contains a quasi-line, ie a smooth rational curve whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: There is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification X' which is strongly-rational We prove that strongly-rational …

关于α-和δ-不变量的准单变量性

On the quasi-monomiality of the $α$- and $δ$-invariants

本文研究了在代数闭域上，对于任何准正的有理极子L，α和δ不变量通过准单变量计算，无需假设基础域的不可数性。