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Homogeneous Khovanskii bases and MUVAK bases

Johannes Schmitt

TL;DR

This work extends the theory of Khovanskii bases by introducing Δ-homogeneous Khovanskii bases and their multi-valuation generalization, MUVAK bases, and provides algorithms for computing them. It establishes a robust framework connecting valuations, graded structures, and initial forms, and demonstrates termination under suitable finite-dimensional conditions. The authors apply these concepts to Cox rings of minimal models for quotient singularities, showing how Ab(G)^∨-homogeneous MUVAK bases generate the Cox ring, and compare their multi-valuation method favorably against Yamagishi's algorithm. Overall, the paper advances algorithmic tools for a broad class of graded algebras and their geometric applications, including explicit computations of Cox rings.

Abstract

In 2019, Kaveh and Manon introduced Khovanskii bases as a special 'Gröbner-like' generating system of an algebra. We extend their work by considering an arbitrary grading on the algebra and propose a definition for a 'homogeneous Khovanskii basis' that respects this grading. We generalize Khovanskii bases further by taking multiple valuations into account (MUVAK bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a quotient singularity. Our algorithm is an improvement of an algorithm by Yamagishi in this situation.

Homogeneous Khovanskii bases and MUVAK bases

TL;DR

This work extends the theory of Khovanskii bases by introducing Δ-homogeneous Khovanskii bases and their multi-valuation generalization, MUVAK bases, and provides algorithms for computing them. It establishes a robust framework connecting valuations, graded structures, and initial forms, and demonstrates termination under suitable finite-dimensional conditions. The authors apply these concepts to Cox rings of minimal models for quotient singularities, showing how Ab(G)^∨-homogeneous MUVAK bases generate the Cox ring, and compare their multi-valuation method favorably against Yamagishi's algorithm. Overall, the paper advances algorithmic tools for a broad class of graded algebras and their geometric applications, including explicit computations of Cox rings.

Abstract

In 2019, Kaveh and Manon introduced Khovanskii bases as a special 'Gröbner-like' generating system of an algebra. We extend their work by considering an arbitrary grading on the algebra and propose a definition for a 'homogeneous Khovanskii basis' that respects this grading. We generalize Khovanskii bases further by taking multiple valuations into account (MUVAK bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a quotient singularity. Our algorithm is an improvement of an algorithm by Yamagishi in this situation.
Paper Structure (14 sections, 23 theorems, 60 equations, 1 table, 3 algorithms)

This paper contains 14 sections, 23 theorems, 60 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Homogeneous Khovanskii bases
  3. Setting
  4. The homogeneous subduction algorithm
  5. Computing homogeneous Khovanskii bases
  6. A generalization to multiple valuations
  7. MUVAK bases
  8. Negative homogenization
  9. Computing homogeneous MUVAK bases
  10. Examples
  11. Cox rings of minimal models of quotient singularities
  12. Quotient singularities
  13. Comparison to Yamagishi's algorithm
  14. A generalization of Bayer's method

Key Result

Proposition 2.11

Algorithm alg:homsubduc is correct assuming it terminates.

Theorems & Definitions (72)

  • Definition 2.1: Valuations
  • Definition 2.2: Initial forms
  • Definition 2.4: Homogeneous Khovanskii bases
  • Example 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.10
  • Proposition 2.11
  • proof
  • ...and 62 more