Generic curves and non-coprime Catalans

TL;DR

This paper computes Poincaré polynomials for certain plane curve singularities and links them to non-coprime q,t-Catalan numbers, confirming a related conjecture.

Contribution

It extends the understanding of compactified Jacobians and their connection to non-coprime q,t-Catalan numbers, confirming a conjecture for specific singularities.

Findings

Computed Poincaré polynomials for singularities with Puiseaux exponents (nd, md, md+1)

Established a relation between these polynomials and non-coprime q,t-Catalan numbers

Confirmed Cherednik and Danilenko's conjecture for these curves

Abstract

We compute the Poincar\'e polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents $(nd,md,md+1)$, and relate them to the combinatorics of $q,t$-Catalan numbers in the non-coprime case. We also confirm a conjecture of Cherednik and Danilenko for such curves.