Mathematics

In this talk, we introduce our prototype of "moduli detector" via derived categories.  Generic elliptic 3-folds are elliptic fibration with certain properties shared by general members of many families of elliptic 3-folds. The smooth part of a generic elliptic fibration is classified by the Tate--Shafarevich group. Recently, we prove a reconstruction theorem for generic elliptic 3-folds with the same relative Jacobian. Namely, two generic elliptic 3-folds are linear derived-equivalent if and only if one of them is a fine relative moduli space of the other of semistable sheaves of rank 1 with an appropriate degree. It follows that the quotient set of the Tate--Shafarevich group by a natural equivalence relation classifies derived categories of generic elliptic 3-folds with the same relative Jacobian up to linear equivalence.   The reconstruction problem will be more interesting for Calabi--Yau 3-folds from the viewpoint of mirror symmetry. In this case, we prove further that linear derived-equivalent generic elliptic 3-folds share the same relative Jacobian. Hence via a linear derived equivalence of generic elliptic Calabi--Yau 3-folds one always find a fine relative moduli space of semistable sheaves of rank 1 with an appropriate degree. Here, the quotient set of the Tate--Shafarevich gets reduced to the quotient set of the Brauer group of the relative Jacobian by the induced equivalence relation, to classify derived categories of generic elliptic Calabi--Yau 3-folds up to linear equivalence.   Considering the goal of this series of seminors, I will try to show how birational geometry plays a role in our study.