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Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$

Jun-ichi Matsushita

Abstract

Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let $Γ$ be a subgroup of $\operatorname{Aut}(K)$ and let $Γ$ act on $P^{n}(K)$ by $σ((x_{i})_{0\leq i\leq n})=(σ(x_{i}))_{0\leq i\leq n}$ for $σ\inΓ$ and $(x_{i})_{0\leq i\leq n}\in P^{n}(K)$. In this paper, we ask `how much' unstable $V$ is under $Γ$ by asking how much higher (or lower) dimension the join (or the meet) of $σ(V)$ ($σ\inΓ$) has than $V$, and answer it in terms of the Plücker coordinates of $V$ and the invariant field $k$ of $Γ$, through presenting dual `irrationality' measures of $V$ over $k$.

Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$

Abstract

Let be a commutative field and let be a subspace of . Let be a subgroup of and let act on by for and . In this paper, we ask `how much' unstable is under by asking how much higher (or lower) dimension the join (or the meet) of () has than , and answer it in terms of the Plücker coordinates of and the invariant field of , through presenting dual `irrationality' measures of over .

Paper Structure

This paper contains 3 sections, 2 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Duality of (\ref{['equality1']}) and (\ref{['equality2']})
  3. Proof of (\ref{['equality2']})

Key Result

Proposition 1

For every $m$-dimensional subspace $X$ of $P^{n}(K)$, for every permutation $j_{0}\cdots j_{n}$ of $0\cdots n$ such that $X_{j_{0}\cdots j_{m}}\neq0$, the right-hand side of (irrq) is equal to the expression obtained by replacing $X$ by $X^{\perp}$, $j_{0}\cdots j_{n}$ by $j_{m+1}\cdots j_{n}j_{0}\c where $\left[ X^{\perp}\right] _{j_{m+1}\cdots j_{n}j_{0}\cdots j_{m}}^{\ast}$ denotes the linear

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4