[Paper Review] On the multiplicity of terminal singularities on threefolds

Nobuyuki Kakimi|ArXiv.org|Apr 17, 2000

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes explicit formulas for the multiplicity and embedding dimension of terminal singularities on threefolds over ℂ, proving that multₚX ≤ indexₚX + 2. Using this bound, it improves the boundedness result for terminal weak ℚ-Fano 3-folds and extends Fujita's freeness conditions to nonhypersurface terminal singularities.

ABSTRACT

We give the multiplicity of terminal singularities on threefolds by simple calculation. Then we obtain the best inequalities for the multiplicity and the index. By using this, we can improve the boundedness number of terminal weak Q-Fano 3-folds in [KMMT, Theorem 1.2]. Furthermore, we can extended [K, Theorem 3.6] for Fujita freeness conditions to nonhypersurface terminal singularities.

Motivation & Objective

• To determine the exact multiplicity and embedding dimension of terminal singularities on threefolds.
• To establish sharp inequalities between multiplicity and index for terminal singularities.
• To improve the boundedness number for terminal weak ℚ-Fano 3-folds using tighter multiplicity bounds.
• To extend Fujita's freeness conditions to nonhypersurface terminal singularities in dimension 3.
• To provide explicit formulas for the Hilbert-Samuel multiplicity of the local ring at terminal singular points.

Proposed method

• Derives the dimension of the graded pieces of the local ring at a terminal singularity: dim(mₚ^k / mₚ^{k+1}) = multₚX · k(k+1)/2 + k + 1.
• Uses Mori’s classification of terminal 3-fold singularities to analyze each type (cA/r, cAx/4, cAx/2, cD/2, cD/3, cE/2).
• Applies the continued fraction algorithm to compute multiplicity for quotient singularities of type ℂ³/ℤᵣ(a,−a,1), yielding multₚX = ⌊r₀/r₁⌋ + ⌊r₁/r₂⌋ + ⋯ + ⌊rₙ₋₁/rₙ⌋ + 2.
• Employs Lemma 3.2 (a volume bound via nef and big divisors and multiplicity) to bound the anti-canonical volume.
• Replaces the previous bound in [KMMT] with the improved multiplicity bound multₚX ≤ indexₚX + 2 in the volume estimate.
• Adapts the proof strategy of [K, Theorem 3.6] by incorporating the new multiplicity bound to extend Fujita freeness to nonhypersurface terminal singularities.

Experimental results

Research questions

• RQ1 What is the exact formula for the Hilbert-Samuel multiplicity of the local ring at a terminal 3-fold singularity?
• RQ2 What is the sharp upper bound of the multiplicity of a terminal singularity in terms of its index?
• RQ3 Can the boundedness number for terminal weak ℚ-Fano 3-folds be improved using tighter multiplicity bounds?
• RQ4 Can Fujita’s freeness condition be extended to nonhypersurface terminal singularities in dimension 3?
• RQ5 How does the embedding dimension of a terminal 3-fold singularity relate to its multiplicity and index?

Key findings

• The multiplicity of a terminal 3-fold singularity satisfies multₚX ≤ indexₚX + 2, with equality possible when indexₚX ≥ 2.
• For a terminal singularity, the embedding dimension is emb.dimₚX = multₚX + 2.
• The multiplicity of a quotient singularity ℂ³/ℤᵣ(a,−a,1) is given by the sum of floor divisions in the continued fraction expansion of r/a, plus 2.
• The improved bound leads to the volume bound (−K_X)³ ≤ 6³·(2 + 24!) for terminal weak ℚ-Fano 3-folds with no divisorial contraction in the anti-canonical morphism.
• The Gorenstein index I(X) divides 24! for terminal weak ℚ-Fano 3-folds, as −K_X·c₂(X) ≥ 0.
• Fujita’s freeness condition is extended to nonhypersurface terminal singularities: if L is ample and satisfies certain volume conditions, then |K_X + L| is free at the singular point.