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

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.

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.