Arc Spaces and Rogers-Ramanujan Identities

TL;DR

This paper introduces a novel algebraic geometric approach to Rogers-Ramanujan identities using arc spaces, connecting singularity theory with combinatorics through Hilbert-Poincaré series and partition generating functions.

Contribution

It establishes a new method linking arc spaces in algebraic geometry to classical partition identities, providing fresh insights into their combinatorial structure.

Findings

Arc spaces relate to partition generating functions.

Hilbert-Poincaré series encodes combinatorial identities.

New geometric perspective on Rogers-Ramanujan identities.

Abstract

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan Identities. The linking object is the Hilbert-Poincar\'e series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts.