Consider a normal variety $S$, and a ramified double cover $f:\tilde{S}\to S$, $\tilde{S}$ smooth, such that $f$ is étale over the nonsingular locus of $S$. Call $j$ the covering involution.
This is the most complete reference I could find:
https://arxiv.org/pdf/1803.00799.pdf
, Section $2$, but in general I hardly find references about ramification coverings. Here it is said that the structure sheaf of the covering involution may be written as $\mathcal{O}_X\oplus \mathcal{R}$, where $\mathcal{R}$ is a certain reflexive sheaf on $X$, see Lemma $2.6$.
The covering involution acts as the identity on the pullback of the structure sheaf of $X$, and as minus the identity on $\mathcal{R}$.
This leads me to think that the involution $j$ acts as the identity on a class $D$ in the Néron-Severi group of $\tilde{S}$ if and only if $D$ can be written as the pullback through $f$ of a class in the Néron-Severi group of $S$. Anyway I couldn't figure out how to prove it.
Is it correct? Thank you!
Best Answer
No, this is not correct. Take for example an etale double covering $C' \to C$ of smooth projective curves. Then $NS(C') = \mathbb{Z}$ via the degree map, and similarly $NS(C) = \mathbb{Z}$. Furthermore, the involution acts on $NS(C')$ trivially. But the image of the pullback map $NS(C) \to NS(C')$ is the subgroup $2\mathbb{Z} \subset \mathbb{Z} = NS(C')$ of index 2.