In this talk, we introduce our prototype of "moduli detector" via derived categories. Generic elliptic 3folds are elliptic fibration with certain properties shared by general members of many families of elliptic 3folds. The smooth part of a generic elliptic fibration is classified by the TateShafarevich group. Recently, we prove a reconstruction theorem for generic elliptic 3folds with the same relative Jacobian. Namely, two generic elliptic 3folds are linear derivedequivalent 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 TateShafarevich group by a natural equivalence relation classifies derived categories of generic elliptic 3folds with the same relative Jacobian up to linear equivalence. The reconstruction problem will be more interesting for CalabiYau 3folds from the viewpoint of mirror symmetry. In this case, we prove further that linear derivedequivalent generic elliptic 3folds share the same relative Jacobian. Hence via a linear derived equivalence of generic elliptic CalabiYau 3folds one always find a fine relative moduli space of semistable sheaves of rank 1 with an appropriate degree. Here, the quotient set of the TateShafarevich 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 CalabiYau 3folds 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.
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Reduced TateShafarevich group
Research Group:
Speaker:
Hayato Morimura
Schedule:
Friday, May 20, 2022  16:30 to 17:30
Location:
A136
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