Let $X$ be a (smooth) del Pezzo surface over $\mathbb{C}$. Let $\Delta\_0$ be a (smooth irreducible) generic curve in the linear system $|-2K\_X|$. Let $\rho : S \rightarrow X$ be the double cover of $X$ branched over $\Delta\_0$ and let $i$ be the associated involution on $S$. Let $\Delta$ be the branching curve of $\rho$, then $\Delta \in |\rho^\*(-K\_X)|$ is a fixed point for the action of $i^\*$ on the linear system $|\rho^\*(-K\_X)|$.


I would like to know if it is always an **isolated** fixed point for the action of $i^\*$ on $|\rho^\*(-K\_X)|$? I am reading a paper where it is stated (without proof) to be true if $X$ is a del Pezzo surface of degree $2$. I don't know how to prove it in the case of the del Pezzo of degree $2$ and I'd like to know if it may be true for other Del Pezzo surfaces.



Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface
One has
$$
\rho\_\*\mathcal{O}\_S \cong \mathcal{O}\_X \oplus \omega\_X
$$
and the involution of $S$ induces the involution of this sheaf that acts by 1 on the first summand and by $-1$ on the second. Consequently,
$$
\rho\_\*\rho^\*\omega\_X^{-1} \cong \omega\_X^{-1} \oplus \mathcal{O}\_X
$$
and the involution still acts by 1 on the first summand and by $-1$ on the second. Therefore, the fixed locus of the involution in the linear system $|\rho^\*(-K\_X)|$ is
$$
\rho^\*|-K\_X| \sqcup \{\Delta\}
$$
and so $\{\Delta\}$ is indeed isolated in the fixed locus.


