Question

Let $S$ be a non-empty closed subscheme of $P^1_K$, where $K$ is a number field. Assume that the cardinality of $S$ is finite.

Is $S$ closed under the action of the absolute Galois group of the field of rational numbers ?

I would like to know this because a certain theorem I want to apply requires this condition.

Answer 1

If you mean to use the action of the absolute Galois group of $K$, then this is true for closed subschemes of $P^n_K$, pretty much by definition. A closed subscheme of $P^n_K={\rm Proj}K[x_0,...,x_n]$ is described by a homogeneous ideal $I$, and the closed subscheme is equal to $W={\rm Proj}K[x_0,...,x_n]/I$. (More precisely, $W$ embeds naturally into $P_K^n$.) Then ${\rm Gal}(\bar K/K)$ acts on $W$, since the ideal $I$ is Galois invariant. So for example, ${\rm Gal}(\bar K/K)$ clearly acts on the geometric points $W(\bar K)$ of $W$. (But maybe I'm missing some subtlety here.)

But if you really mean to look at the action of ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, then it's certainly false. For example, take $K=\mathbb{Q}(i)$ and $S$ to be the subscheme consisting of the single point $[i,1]\in P^1_K$. (Formally, $S$ is associated to the ideal generated by the polynomial $x-iy$, which defines a closed subscheme of $P_K^1$.) The ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ orbit of $S$ consists of two points.