# On equivalent conjectures for minimal log discrepancies on smooth threefolds

@article{Kawakita2020OnEC, title={On equivalent conjectures for minimal log discrepancies on smooth threefolds}, author={Masayuki Kawakita}, journal={Journal of Algebraic Geometry}, year={2020} }

On smooth varieties, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. In dimension three, we reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.

## 14 Citations

The minimal log discrepancies on a smooth surface in positive characteristic

- MathematicsMathematische Zeitschrift
- 2020

This paper shows that Mustaţǎ–Nakamura’s conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As…

ACC for minimal log discrepancies of exceptional singularities.

- Mathematics
- 2019

We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the…

Divisors computing minimal log discrepancies on lc surfaces

- Mathematics
- 2021

Let (X ∋ x,B) be an lc surface germ. If X ∋ x is klt, we show that there exists a divisor computing the minimal log discrepancy of (X ∋ x,B) that is a Kollár component of X ∋ x. If B 6= 0 or X ∋ x is…

On generalized minimal log discrepancy

- Mathematics
- 2021

We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for…

Inversion of modulo p reduction and a partial descent from characteristic 0 to positive characteristic

- Mathematics
- 2018

In this paper we focus on pairs consisting of the affine $N$-space and multiideals with a positive exponent. We introduce a method "lifting to characteristic 0" which is a kind of the inversion of…

Inversion of adjunction for quotient singularities II: Non-linear actions

- Mathematics
- 2021

We prove the precise inversion of adjunction formula for quotient singularities. As an application, we prove the semi-continuity of minimal log discrepancies for hyperquotient singularities. This…

ACC for local volumes and boundedness of singularities

- Mathematics
- 2020

The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a DCC set. In this…

Inversion of adjunction for quotient singularities

- MathematicsAlgebraic Geometry
- 2022

We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt…

Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies.

- Mathematics
- 2018

In this paper, we show that Shokurov's conjectures on the ACC for $a$-lc thresholds and the ACC for minimal log discrepancies are equivalent in the interval $[0,1)$. That is, the conjecture on ACC…

Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces

- Mathematics
- 2020

Fix a subset $I\subseteq \mathbb R_{>0}$ such that $\gamma=\inf\{ \sum_{i}n_ib_i-1>0 \mid n_i\in \mathbb Z_{\geq 0}, b_i\in I \}>0$. We give a explicit upper bound $\ell(\gamma)\in O(1/\gamma^2)$ as…

## References

SHOWING 1-10 OF 31 REFERENCES

A boundedness conjecture for minimal log discrepancies on a fixed germ

- Mathematics
- 2015

We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor…

Shokurov's ACC conjecture for log canonical thresholds on smooth varieties

- Mathematics
- 2009

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log…

Divisors computing the minimal log discrepancy on a smooth surface

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract We study a divisor computing the minimal log discrepancy on a smooth surface. Such a divisor is obtained by a weighted blow-up. There exists an example of a pair such that any divisor…

Log canonical thresholds on varieties with bounded singularities

- Mathematics
- 2010

We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also…

Ideal-adic semi-continuity of minimal log discrepancies on surfaces

- Mathematics
- 2012

We prove the ideal-adic semi-continuity of minimal log dis crepancies on surfaces. De Fernex, Ein and Mustaţă in [1] after Kollár in [4] prove d the ideal-adic semicontinuity of log canonicity…

A connectedness theorem over the spectrum of a formal power series ring

- Mathematics
- 2014

We study the connectedness of the non-klt locus over the spectrum of a formal power series ring. In dimension 3, we prove the existence and normality of the smallest lc center, and apply it to the…

Discreteness of log discrepancies over log canonical triples on a fixed pair

- Mathematics
- 2012

For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.

Rational And Nearly Rational Varieties

- Mathematics
- 2004

Introduction 1. Examples of rational varieties 2. Cubic surfaces 3. Rational surfaces 4. Nonrationality and reduction modulo p 5. The Noether-Fano method 6. Singularities of pairs 7. Solutions to…

Two two-dimensional terminations

- Mathematics
- 1992

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the…

General elephants of three-fold divisorial contractions

- Mathematics
- 2002

The theory of minimal models has enriched the study of higher-dimensional algebraic geometry; see [10] and [12]. For a variety with mild singularities, this theory produces another variety which…