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

Abstract

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