Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface

Question

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.

Answer 1

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.