Reduced Tate-Shafarevich group

Abstract: We prove a sort of reconstruction theorem for generic elliptic Calabi-Yau $3$-folds in the sense of C\u{a}ld\u{a}raru. From our argument it follows that two generic elliptic Calabi-Yau $3$-folds are derived-equivalent linear over the base if and only if their generic fibers are derived-equivalent. As an application, we give affirmative answers to the conjectures raised by Knapp-Scheidegger-Schimannek. Namely, for each pair of elliptic Calabi-Yau $3$-folds in their list we prove that they share the relative Jacobian and are $P^2$-linear derived-equivalent.