Academic Commons

Theses Doctoral

Brauer class over the Picard scheme of curves

Ma, Qixiao

We study the Brauer classes rising from the obstruction to the existence of tautological line bundles on the Picard scheme of curves. We establish various properties of the Brauer classes for families of smooth curves. We compute the period and index of the Brauer class associated with the universal smooth curve for a fixed genus. We also show such Brauer classes are trivialized when we specialize to certain generalized theta divisors. If we consider the universal totally degenerate curve with a fixed dual graph, using symmetries of the graph, we give bounds on the period and index of the Brauer classes. As a result, we provide some division algebras of prime degree, serving as candidates for the cyclicity problem. As a byproduct, we re-calculate the period and index of the Brauer class for universal smooth genus g curve in an elementary way. We study certain conic associated with the universal totally degenerate curve with a fixed dual graph. We show the associated conic is non-split in some cases. We also study some other related geometric properties of Brauer groups.

Files

  • thumnail for Ma_columbia_0054D_15155.pdf Ma_columbia_0054D_15155.pdf application/pdf 658 KB Download File

More About This Work

Academic Units
Mathematics
Thesis Advisors
de Jong, Aise Johan
Degree
Ph.D., Columbia University
Published Here
April 24, 2019
Academic Commons provides global access to research and scholarship produced at Columbia University, Barnard College, Teachers College, Union Theological Seminary and Jewish Theological Seminary. Academic Commons is managed by the Columbia University Libraries.