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On reduced arc spaces of toric varieties

Ilya Dumanski, Evgeny Feigin, Ievgen Makedonskyi, Igor Makhlin

TL;DR

This work develops a general machinery to describe reduced arc spaces $J^ abla_{ ext{red}}(R(P))$ for affine toric cones by translating arc-relations into spaces of partially symmetric polynomials and current-algebra actions. It builds a dual picture where graded components correspond to modules over algebras of symmetric polynomials, and identifies when these actions are free (favourable polytopes), enabling explicit character computations. The authors provide an inductive construction that yields large families of favourable polytopes (notably simplices, parallelepipeds, Hirzebruch trapezoids) and apply the framework to Veronese–Segre-type embeddings, connecting reduced arc rings with affine Demazure modules and semi-infinite flag variety geometry. The results illuminate structural regularities in reduced arc spaces, give concrete descriptions in classical cases, and link differential-ideal finiteness with representation-theoretic objects, offering a robust toolkit for further exploration at the intersection of toric geometry, singularity theory, and representation theory.

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.

On reduced arc spaces of toric varieties

TL;DR

This work develops a general machinery to describe reduced arc spaces for affine toric cones by translating arc-relations into spaces of partially symmetric polynomials and current-algebra actions. It builds a dual picture where graded components correspond to modules over algebras of symmetric polynomials, and identifies when these actions are free (favourable polytopes), enabling explicit character computations. The authors provide an inductive construction that yields large families of favourable polytopes (notably simplices, parallelepipeds, Hirzebruch trapezoids) and apply the framework to Veronese–Segre-type embeddings, connecting reduced arc rings with affine Demazure modules and semi-infinite flag variety geometry. The results illuminate structural regularities in reduced arc spaces, give concrete descriptions in classical cases, and link differential-ideal finiteness with representation-theoretic objects, offering a robust toolkit for further exploration at the intersection of toric geometry, singularity theory, and representation theory.

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.
Paper Structure (24 sections, 64 theorems, 266 equations)

This paper contains 24 sections, 64 theorems, 266 equations.

Table of Contents

  1. Preliminaries and generalities
  2. Toric varieties
  3. Arc spaces
  4. Derivations of arc rings
  5. Supersymmetric polynomials
  6. Associated graded algebras and initial ideals
  7. Unit cubes
  8. Symmetric polynomials and dual spaces of graded components
  9. Arcs over the polynomial ring
  10. Action of the current Lie algebra on dual components
  11. Arcs and quadratic monomial ideals
  12. Arc rings of toric rings
  13. Graded components of toric rings
  14. Dual of the inclusion map
  15. Current algebra action and $\mathcal{A}$-freeness
  16. ...and 9 more sections

Key Result

Theorem 1

There is a homogeneous element $K \in \Lambda_{\bar{r}}(\mathbf{t})$ such that

Theorems & Definitions (125)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Definition 1.5
  • ...and 115 more