[Paper Review] On minimal log discrepancies on varieties with fixed Gorenstein index

Yusuke Nakamura|arXiv (Cornell University)|Jan 26, 2015

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper establishes the discreteness and ACC (Ascending Chain Condition) for minimal log discrepancies (mld) on three-dimensional canonical pairs with fixed coefficients, by generalizing rationality theorems on log canonical thresholds. It proves that mld values are discrete in ℝ for varieties with fixed Gorenstein index, leading to the ACC for mld in dimension 3 when coefficients are finite, resolving a key case in the minimal model program.

ABSTRACT

We generalize the rationality theorem of the accumulation points of log canonical thresholds which was proved by Hacon, M extsuperscript{c}Kernan, and Xu. Further, we apply the rationality to the ACC problem on the minimal log discrepancies. We study the set of log discrepancies on varieties with fixed Gorenstein index. As a corollary, we prove that the minimal log discrepancies of three-dimensional canonical pairs with fixed coefficients satisfy the ACC.

Motivation & Objective

• To generalize the rationality theorem of log canonical thresholds to varieties with fixed Gorenstein index.
• To investigate the ACC conjecture for minimal log discrepancies in the context of fixed Gorenstein index.
• To prove that the set of minimal log discrepancies is discrete in ℝ for varieties with fixed Gorenstein index.
• To establish the ACC for minimal log discrepancies in three-dimensional canonical pairs with finite coefficient sets.
• To provide a foundation for the boundedness and finiteness of mld sets under Gorenstein index constraints.

Proposed method

• Generalizing the rationality theorem of Hacon–McKernan–Xu to log pairs with fixed Gorenstein index.
• Using induction on the dimension of the ℚ-vector space spanned by the coefficient set I ∪ {1}.
• Applying a perturbation theorem (Theorem 1.6) on irrational coefficients in log canonical pairs, ensuring lc property is preserved under small perturbations.
• Constructing a discrete set of log discrepancies via finite unions of scaled rational sets and discrete subsets.
• Reducing the mld problem to bounded Gorenstein index cases via the generic limit method and canonical singularity classification.
• Using the boundedness of mld in dimension 3 (BDD conjecture) and classification of 3D terminal singularities to bound mld values.

Experimental results

Research questions

• RQ1 Does the set of minimal log discrepancies on varieties with fixed Gorenstein index satisfy the ACC?
• RQ2 Can the rationality theorem on log canonical thresholds be extended to varieties with fixed Gorenstein index?
• RQ3 Is the set of log discrepancies discrete for log pairs with fixed Gorenstein index and finite coefficient set?
• RQ4 What is the structure of minimal log discrepancies in three-dimensional canonical pairs with fixed coefficients?
• RQ5 How does the Gorenstein index constrain the accumulation points of minimal log discrepancies?

Key findings

• The set $ B(d,r,I) $ of log discrepancies $ a_E(X,rak{a}) $ is discrete in $ bR $ for $ d $-dimensional log canonical pairs $ (X,rak{a}) $ with $ rK_X $ Cartier and $ rak{a} $ having coefficients in a finite set $ I $.
• The set $ A'(d,r,I) $ of minimal log discrepancies at closed points is discrete in $ bR $, though not necessarily finite without boundedness.
• For three-dimensional canonical pairs with finite coefficient set $ I o [0,1] $, the set $ A_{ ext{can}}(3,I) $ satisfies the ACC, with 1 as its only accumulation point.
• The minimal log discrepancy of a 3D terminal singularity is either $ 1 + 1/r $ (for Gorenstein index $ r $) or 3, which helps bound the mld values.
• When $ ext{mld}_x(X, riangle) o a > 1 $, the Gorenstein index of the ambient variety is bounded by $ loor{1/(a-1)} $, enabling finite classification.
• The proof relies on reduction to bounded Gorenstein index cases via canonical models and the classification of 3D terminal singularities, showing $ A_{ ext{can}}(3,I) o igcup_{l o loor{1/(a-1)}} A'(3,l, I/l) $, which is finite by Corollary 1.3.