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Strong Watanabe-Yoshida conjecture for Complete Intersections

Joel Castillo-Rey

TL;DR

The paper settles the Strong Watanabe–Yoshida conjecture for complete intersections over all positive characteristics by harnessing the Han–Monsky representation ring to compare Hilbert–Kunz multiplicities and functions. It develops a detailed, characteristic-sensitive computation framework in the Han–Monsky ring (including δ/λ and σ-bases) to obtain explicit formulas for A1 and A2 singularities in characteristics 2 and 3, and uses dense upper semi-continuity plus Weierstrass preparation to lift hypersurface results to the complete-intersection setting. The main results show e_HWK(S_{p,d}) > e_HWK(R_{p,d}) whenever R is not isomorphic to the quadric model R_{p,d}; consequently, the minimal HK value is achieved only for these quadric cases, proving the conjecture in full for complete intersections. In addition, the paper provides explicit HK functions for A1 and A2 in p=2,3 and verifies (or refines) related bounds in low dimensions and multiplicities, yielding a characteristic-free perspective up to dimension 7. Overall, the work advances the understanding of Hilbert–Kunz behavior across complete intersections and consolidates a framework for characteristic-dependent HK computations with broad implications for singularity theory in positive characteristic.

Abstract

In this paper, we prove the strong form of the Watanabe-Yoshida conjecture for complete intersection singularities in every positive characteristic. In characteristics 2 and 3, we explicitly compute the Hilbert-Kunz functions of the A1 and A2 singularities.

Strong Watanabe-Yoshida conjecture for Complete Intersections

TL;DR

The paper settles the Strong Watanabe–Yoshida conjecture for complete intersections over all positive characteristics by harnessing the Han–Monsky representation ring to compare Hilbert–Kunz multiplicities and functions. It develops a detailed, characteristic-sensitive computation framework in the Han–Monsky ring (including δ/λ and σ-bases) to obtain explicit formulas for A1 and A2 singularities in characteristics 2 and 3, and uses dense upper semi-continuity plus Weierstrass preparation to lift hypersurface results to the complete-intersection setting. The main results show e_HWK(S_{p,d}) > e_HWK(R_{p,d}) whenever R is not isomorphic to the quadric model R_{p,d}; consequently, the minimal HK value is achieved only for these quadric cases, proving the conjecture in full for complete intersections. In addition, the paper provides explicit HK functions for A1 and A2 in p=2,3 and verifies (or refines) related bounds in low dimensions and multiplicities, yielding a characteristic-free perspective up to dimension 7. Overall, the work advances the understanding of Hilbert–Kunz behavior across complete intersections and consolidates a framework for characteristic-dependent HK computations with broad implications for singularity theory in positive characteristic.

Abstract

In this paper, we prove the strong form of the Watanabe-Yoshida conjecture for complete intersection singularities in every positive characteristic. In characteristics 2 and 3, we explicitly compute the Hilbert-Kunz functions of the A1 and A2 singularities.

Paper Structure

This paper contains 18 sections, 59 theorems, 128 equations, 3 tables.

Table of Contents

  1. The Han–Monsky ring and the Han–Monsky algorithm
  2. Definitions
  3. The Han–Monsky ring in characteristic 2
  4. The Han–Monsky algorithm
  5. On the Hilbert–Kunz multiplicity of the A1 and A2 singularities, p gt 3
  6. The Han–Monsky algorithm for the A1 and A2 singularities, p gt 3
  7. The coefficients of the powers of two k-objects
  8. The main inequality, p ge 3
  9. On the Hilbert–Kunz function of the A1 and A2 singularities, p23
  10. The A1 singularity in characteristic 2
  11. The A2 singularity in characteristic 2
  12. The A1 and A2 singularity in characteristic 3
  13. The main inequality, p23
  14. The Strong Watanabe–Yoshida conjecture for complete intersections
  15. Dense upper semi-continuity and Weierstrass' Preparation Lemma
  16. ...and 3 more sections

Key Result

Theorem 1

With the same notation as in the conjecture above, the following hold.

Theorems & Definitions (135)

  • Conjecture 1: wy05, yos
  • Theorem 1
  • Theorem 2: Enescu–Shimomoto, enshi, Theorems 3.4 and 4.6
  • Theorem A: Corollary \ref{['maintheoremA']}
  • Theorem B: Theorems \ref{['th: HK_function_A_1_char_2']}, \ref{['th: HK_function_A_2_char_2']}, \ref{['th: eHK_A_1_and_A_2_in_char_3']}
  • Theorem C: Theorem \ref{['th: watanabe_yoshida_low_dimension']}, \ref{['cor: watanabe_yoshida_low_multiplicity']}
  • Definition 1.1: $k$-object
  • Proposition 1.2: Definition 1.3, hanmonsky
  • Example 1.3
  • Definition 1.4: Han–Monsky ring
  • ...and 125 more