# What is an invariant embedding?

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 :).

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 $$i$$