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A note on Bridgeland stability conditions and Catalan numbers

Jason Lo and Karissa Wong

Vol. 15 (2022), No. 3, 427–432
Abstract

We describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface and the action of its autoequivalence group on its Bridgeland stability manifold. In solving an equation involving this group action, the generating function of Catalan numbers arises, allowing us to use asymptotic estimates of Catalan numbers to arrive at a bound for the solution set.

Keywords
Bridgeland stability, Catalan numbers, autoequivalence
Mathematical Subject Classification
Primary: 14F08, 14J27
Secondary: 05A15, 05E14
Milestones
Received: 21 April 2021
Revised: 9 October 2021
Accepted: 30 October 2021
Published: 2 December 2022

Communicated by Ravi Vakil
Authors
Jason Lo
Department of Mathematics
California State University
Northridge, CA
United States
Karissa Wong
Department of Mechanical Engineering
University of California
Berkeley, CA
United States