# On minimal log discrepancies

@article{Ambro1999OnML, title={On minimal log discrepancies}, author={Florin Ambro}, journal={Mathematical Research Letters}, year={1999}, volume={6}, pages={573-580} }

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 function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.

## 59 Citations

Minimal log discrepancies in positive characteristic

- MathematicsCommunications in Algebra
- 2021

Abstract We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log…

Minimal log discrepancies of regularity one

- Mathematics
- 2021

In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove…

On semi-continuity problems for minimal log discrepancies

- Mathematics
- 2013

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…

On minimal log discrepancies on varieties with fixed Gorenstein index

- Mathematics
- 2015

We generalize the rationality theorem of the accumulation points of log canonical thresholds which was proved by Hacon, M\textsuperscript{c}Kernan, and Xu. Further, we apply the rationality to the…

LETTERS OF A BI-RATIONALIST IV. Geometry of log flips

- Mathematics
- 2002

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the…

Boundedness of $\epsilon$-log Canonical Complements on Surfaces

- Mathematics
- 2004

We prove a conjecture due to V.V. Shokurov on the boundedness of $\epsilon$-log canonical complements on surfaces. As an application we give a new proof to the boundedness of weak log Fano surfaces.

The ACC Conjecture for minimal log discrepancies on a fixed germ

- Mathematics
- 2015

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

On minimal log discrepancies and kollár components

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2021

In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that
$d$
-dimensional
$a$
-log canonical singularities with standard…

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…

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…

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