A note on Bridgeland stability conditions and Catalan numbers

TL;DR

This paper explores a problem in algebraic geometry involving Bridgeland stability conditions on elliptic surfaces, where Catalan numbers naturally appear in the analysis, leading to bounds on solutions.

Contribution

It reveals a novel connection between Bridgeland stability conditions and Catalan numbers, providing new bounds in the context of derived categories on elliptic surfaces.

Findings

Catalan numbers appear in the solution to a stability condition problem.

Asymptotic estimates of Catalan numbers yield bounds on the solution set.

The work links combinatorial sequences with algebraic geometry concepts.

Abstract

In this short note, 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.