# 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} }

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

- MathematicsJournal 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

- Mathematics
- 2020

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

- Mathematics
- 2021

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

- 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…

Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications

- Mathematics
- 2017

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

- MathematicsEuropean Journal of Mathematics
- 2021

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…

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