Intersection numbers on fibrations and Catalan numbers

TL;DR

This paper explains the appearance of Catalan numbers in the context of elliptic fibrations by linking them to specific equations in the Chow ring, shedding light on their geometric significance.

Contribution

It identifies a class of Chow ring equations in fibrations whose solutions inherently involve Catalan numbers, revealing a new geometric connection.

Findings

Catalan numbers arise in Chow ring equations of fibrations.

The results explain the appearance of Catalan numbers in autoequivalence group actions.

Provides a geometric interpretation of Catalan numbers in algebraic geometry.

Abstract

On an elliptic surface or threefold, Catalan numbers appear when one tries to compute the autoequivalence group action on the Bridgeland stability manifold. We explain why this happens by identifying a class of equations in the Chow ring of a fibration, where the solutions always involve Catalan numbers.