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Toricness and smoothness criteria for spherical varieties

Giuliano Gagliardi, Johannes Hofscheier, Heath Pearson

TL;DR

The paper develops a purely combinatorial, skeleton-based criterion to detect toric structure and smoothness along G-orbits for complete spherical varieties. It defines the p-tilde invariant wp-tilde and shows wp-tilde(X) \ge 0, with equality exactly for toric varieties, thereby providing a numerical toricity test from the spherical skeleton. The main technique reduces to the Q-Gorenstein setting via a Gorenstein algorithm that preserves wp-tilde and then to MF-space theory, enabling a complete verification by known classifications and explicit computations. The paper also derives a practical smoothness criterion along any G-orbit and offers concrete examples, including a singular locally factorial embedding and a smooth conic parametrization, illustrating the scope and limitations of the approach within spherical geometry.

Abstract

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a coarse ''subset'' of the Luna-Vust data of a spherical variety. Our smoothness criterion improves upon classical criteria by removing the dependency on external reference tables.

Toricness and smoothness criteria for spherical varieties

TL;DR

The paper develops a purely combinatorial, skeleton-based criterion to detect toric structure and smoothness along G-orbits for complete spherical varieties. It defines the p-tilde invariant wp-tilde and shows wp-tilde(X) \ge 0, with equality exactly for toric varieties, thereby providing a numerical toricity test from the spherical skeleton. The main technique reduces to the Q-Gorenstein setting via a Gorenstein algorithm that preserves wp-tilde and then to MF-space theory, enabling a complete verification by known classifications and explicit computations. The paper also derives a practical smoothness criterion along any G-orbit and offers concrete examples, including a singular locally factorial embedding and a smooth conic parametrization, illustrating the scope and limitations of the approach within spherical geometry.

Abstract

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a coarse ''subset'' of the Luna-Vust data of a spherical variety. Our smoothness criterion improves upon classical criteria by removing the dependency on external reference tables.
Paper Structure (12 sections, 26 theorems, 39 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 26 theorems, 39 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Technique of the proof
  4. Defining the p-tilde function
  5. The inequality in Theorem \ref{['main thm']}
  6. Proof of Algorithm \ref{['Gorenstein algorithm']}
  7. A colored analogue of star subdivisions
  8. Proving Algorithm \ref{['Gorenstein algorithm']}
  9. Toric varieties and P
  10. Examples
  11. p for the indecomposable saturated multiplicity free spaces.
  12. Explicit values of vartheta_argmax

Key Result

Theorem 1.3

Let $X$ be a complete spherical variety, then with equality if and only if $X$ is isomorphic to a toric variety.

Figures (1)

  • Figure 1: Illustration of \ref{['rem:why-include-abstr-faces']}.

Theorems & Definitions (75)

  • Definition 1.1: GH15
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4: GH15
  • Theorem 1.5
  • Theorem 1.6: GHPsphericalMukai
  • Definition 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.2
  • ...and 65 more