Two arithmetic relations of a subspace of $P^{n}(K)$ and a subgroup of $\operatorname{Aut}(K)$

Jun-ichi Matsushita

Published 2016-03-04Version 1

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 define the action of $\Gamma$ on $P^{n}(K)$ by letting $\sigma((x_{i})_{0\leqslant i\leqslant n})$ be $(\sigma(x_{i}))_{0\leqslant i\leqslant n}$ for $\sigma\in\Gamma$, $(x_{i})_{0\leqslant i\leqslant n}\in P^{n}(K)$. In this paper, using two arithmetic notions defined in terms of the Pl\"{u}cker coordinates of $V$ and the invariant field of $\Gamma$, we answer the questions: By what number is the dimension of the join (resp. meet) of $\sigma(V)$ ($\sigma\in\Gamma$) greater (resp. less) than the dimension of $V$?