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Let $S$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $0$, equipped with a regular involution $\iota$, i.e., with an action of the order two (cyclic) group $G=\{id,\iota\}$ generated by $\iota$.

Let $D$ be a very ample $\iota$-invariant divisor on $S$.

Let $\Sigma$ be the complete linear system of the divisor $D$ and let $d=\dim(\Sigma)$ be the dimension of $\Sigma$.

Let $\phi_{\Sigma}:S\rightarrow \mathbb{P}^d,$ be the closed embedding of the surface $S$ into the projective space $\mathbb{P}^d$ induced by $\Sigma$.

Recall that we can identify $\Sigma= (\mathbb{P}^d)^*$, where $(\mathbb{P}^d)^*$ is the dual projective space of $\mathbb{P}^d$ parametrizing hyperplanes in $\mathbb{P}^d$.

In the paper "https://arxiv.org/abs/1704.04187v2" page 16 says: Since $D$ is $\iota$-invariant, the group $G=\{id, \iota\}$ acts on $\Sigma= (\mathbb{P}^d)^*$ and then also on $\mathbb{P}^d$ such that the embedding $S\hookrightarrow \mathbb{P}^d$ is $\iota$-invariant.

So, I would like to know what does it mean that "the embedding $S\hookrightarrow \mathbb{P}^d$ is $\iota$-invariant" in this context. Any reference also will be very helpful :).

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I think first observe that $i$ acts as $D \mapsto i^*D $ and invariance means that $i^*D=D$ so sections of D are invariant under this pullback. Then $\mathcal{O}_{\mathbf{P}^n}(1)|_S =D$ where the global sections of the former give you the coordinates of $S$ in the projective space via this embedding, which are thus invariant under $i$

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