[Paper Review] Gap conjecture for 3-dimensional canonical thresholds

Yuri Prokhorov|arXiv (Cornell University)|Jun 24, 2008

Polynomial and algebraic computation4 references3 citations

TL;DR

This paper proves that the canonical threshold of a 3-dimensional terminal singularity with a Weil divisor lies at most at 5/6, establishing a sharp gap in the threshold values: no canonical thresholds exist in the interval (5/6, 1). The proof uses weighted blowups and classification of terminal singularities, showing that thresholds exceeding 5/6 violate the canonical condition, with the maximal threshold being exactly 4/5 for singular points.

ABSTRACT

We prove that the interval $(5/6, 1)$ contains no 3-dimensional canonical thresholds.

Motivation & Objective

• To prove the gap conjecture for 3-dimensional canonical thresholds, specifically that the interval (5/6, 1) contains no such thresholds.
• To establish the precise value of the canonical threshold gap for dimension 3, showing ε³^can = 1/6.
• To extend the result to terminal singularities, proving that if (X∋P) is singular, then ct(X,S) ≤ 4/5.
• To resolve Conjecture 1.2 for n=3 by showing that the infimum of thresholds not equal to 1 is bounded away from 1 by 1/6.
• To use weighted blowup techniques and classification of terminal singularities to analyze discrepancies and threshold bounds.

Proposed method

• Apply weighted blowups with specific weights to analyze discrepancies of divisors over terminal singularities.
• Use the formula a(G, K_X + cS) = |α| - 1 - v_α(φ) - c v_α(ψ) to compute discrepancies and bound c.
• Classify terminal singularities in dimension 3 via their local analytic structure, including Gorenstein and non-Gorenstein cases.
• Use index-one covers to reduce non-Gorenstein cases to the Gorenstein setting, preserving canonical properties.
• Apply the criterion that if (X,S) is not canonical, then the pair (S,P) cannot be Du Val, derived from Rees' theorem.
• Analyze the structure of the exceptional divisor and its components under blowups to identify admissible weights and compute discrepancies.

Experimental results

Research questions

• RQ1 Does the set of 3-dimensional canonical thresholds satisfy the ascending chain condition, and is there a gap above 5/6?
• RQ2 What is the maximal possible canonical threshold for a 3-dimensional terminal singularity with a Weil divisor?
• RQ3 Can the canonical threshold exceed 5/6 for any 3-fold terminal singularity?
• RQ4 What is the precise value of ε³^can = 1 - sup(T^can_3 ⋯ {1})?
• RQ5 Under what conditions does the pair (X,S) fail to be canonical, and how does this constrain the threshold c?

Key findings

• The interval (5/6, 1) contains no 3-dimensional canonical thresholds, proving the gap conjecture for n=3.
• The canonical threshold of any 3-dimensional terminal singularity with a Weil divisor satisfies ct(X,S) ≤ 5/6.
• For singular terminal points, the threshold is bounded by 4/5, and this bound is sharp, as shown by a construction with ct = 4/5.
• The maximal threshold ε³^can = 1/6 is achieved, confirming that the gap is exactly 1/6.
• In the non-Gorenstein case, if ct(X,S) > 1/2, then K_X + S ∼ 0, which leads to a contradiction unless S is Du Val, which is impossible.
• The proof establishes that the pair (S,P) cannot be Du Val, a key obstruction used to rule out high thresholds.