On reduced arc spaces of toric varieties

TL;DR

This paper develops a general framework for describing the reduced arc spaces of affine cones over toric varieties, highlighting their connections to representation theory and combinatorics.

Contribution

It introduces new methods for analyzing reduced arc spaces of toric varieties and applies them to classical cases, revealing links to current algebra representations.

Findings

Reduced arc spaces are generally non-reduced but can be effectively described using new machinery.

Applications to classical cases demonstrate the utility of the developed techniques.

Connections with representation theory of current algebras are explored.

Abstract

An arc space of an affine cone over a projective toric variety is known to be non-reduced in general. It was demonstrated recently that the reduced scheme structure is worth studying due to various connections with representation theory and combinatorics. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.