# A boundedness conjecture for minimal log discrepancies on a fixed germ

@article{Musta2015ABC,
title={A boundedness conjecture for minimal log discrepancies on a fixed germ},
author={Mircea Mustaţǎ and Yusuke Nakamura},
journal={arXiv: Algebraic Geometry},
year={2015}
}
• Published 3 February 2015
• Mathematics
• arXiv: Algebraic Geometry
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 E over X that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N. We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture.
17 Citations
On equivalent conjectures for minimal log discrepancies on smooth threefolds
• M. Kawakita
• Mathematics
Journal of Algebraic Geometry
• 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
Small quotient minimal log discrepancies
We prove that for each positive integer $n$ there exists a positive number $\epsilon_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval
A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds
We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general" real ideal. We show that the minimal log discrepancy (“mld" for short) of every such a pair is
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
The minimal log discrepancies on a smooth surface in positive characteristic
• S. Ishii
• Mathematics
Mathematische 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
Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications
We study singularities in arbitrary characteristic. We propose finite determination conjecture for Mather–Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture
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
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
Characterization of two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove

## References

SHOWING 1-10 OF 11 REFERENCES
On minimal log discrepancies
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous
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
Limits of log canonical thresholds
• Mathematics
• 2007
Let Tn denote the set of log canonical thresholds of pairs (X, Y ), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every
Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
Summary.This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spaces
Jet schemes, log discrepancies and inversion of adjunction
• Mathematics
• 2003
Singularities play a key role in the Minimal Model Program. In this paper we show how some of the open problems in this area can be approached using jet schemes. Let (X, Y ) be a pair, where X is a
A connectedness theorem over the spectrum of a formal power series ring
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
On semi-continuity problems for minimal log discrepancies
We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne-Mumford stacks. Using this property, we also prove the
Which powers of holomorphic functions are integrable
We show that every limit of log canonical thresholds of n-variable functions is also a log canonical threshold of an (n-1)-variable function.
Ideal-adic semi-continuity problem for minimal log discrepancies
We discuss the ideal-adic semi-continuity problem for minimal log discrepancies by Mustaţă. We study the purely log terminal case, and prove the semi-continuity of minimal log discrepancies when a
ACC for log canonical thresholds
• Physics, Mathematics
Annals of Mathematics
• 2014
We show that log canonical thresholds satisfy the ACC.