Of course you are both right. It's embarrassing how I messed it out in my mind! Thanks anyway for caring to show me I was nonsense.

Of course you are right! I was mistaken by a comment on another question, my apologies.

@Donu: yes, indeed! anyway my $S$ is projective.

OK I'll add the hypotheses I care about: X is a smooth surface

sure for curves it is fine, yeah

You can reconstruct the genus 2 curve by taking the double cover of the Theta divisor (which is a conic in the singular Kummer - namely the intersection of the Kummer with one of the tropes) ramified along the 6 nodes contained therein.

Just one question @Jason, is it clear that the total space of $Q$ is rational?

this is really brilliant, thank you