# Two two-dimensional terminations

@article{Alexeev1992TwoTT, title={Two two-dimensional terminations}, author={Valery Alexeev}, journal={Duke Mathematical Journal}, year={1992}, volume={69}, pages={527-545} }

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 interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates. Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of…

## 75 Citations

Boundedness of ǫ-lc Complements on Surfaces

- Mathematics
- 2008

Contents 1 Boundedness of ǫ-log canonical complements on surfaces 2 1. 1 Boundedness of ǫ-log canonical complements on surfaces 1.1 Introduction The concept of complement was introduced and studied…

On a conjecture of Shokurov: characterization of toric varieties

- Mathematics
- 2000

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let $(X,D=\sum d_iD_i)$ be a three-dimensional log variety such that $K_X+D$ is…

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…

Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram Method)

- Mathematics
- 1994

In our preprint: "Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces", Preprint Max-Planck-Institut f\"ur Mathematik, Bonn,…

Bounding singular surfaces via Chern numbers

- Mathematics
- 2017

We prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and Chern numbers. As an application, we prove that given $$R\in…

Mld's vs thresholds and flips

- Mathematics
- 2006

Abstract Minimal log discrepancies (mld's) are related not only to termination of log flips [Shokurov, Algebr. Geom. Metody 246: 328–351, (2004)] but also to the ascending chain condition (ACC) of…

Descending chain condition for stringy invariants (Higher Dimensional Algebraic Geometry)

- Mathematics
- 2011

It is known that the degree of the stringy \mathrm{E}‐function of a \log terminal singularity is related to the minimal log discrepancy, and the minimal \log discrepancies of certain classes of…

On numerical nonvanishing for generalized log canonical pairs.

- Mathematics
- 2020

The nonvanishing conjecture for projective log canonical pairs plays a key role in the minimal model program of higher dimensional algebraic geometry. The numerical nonvanishing conjecture considered…

Log canonical $3$-fold complements

- Mathematics
- 2019

We expand the theory of log canonical $3$-fold complements. More precisely, fix a set $\Lambda \subset \mathbb{Q}$ satisfying the descending chain condition with $\overline{\Lambda} \subset…

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…

## References

SHOWING 1-10 OF 16 REFERENCES

Threefolds whose canonical bundles are not numerically effective.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1980

A characterization of an arbitrary nonsingular projective 3-fold whose canonical bundle is not numerically effective and which contains an exceptional divisor of several types, which is classified explicitly.

ON THE CLOSED CONE OF CURVES OF ALGEBRAIC 3-FOLDS

- Mathematics
- 1985

In this paper the author establishes, under natural conditions, the local polyhedrality of the closed cone of curves of a three-dimensional algebraic variety in the part that is negative with respect…

FRACTIONAL INDICES OF LOG DEL PEZZO SURFACES

- Mathematics
- 1989

The fractional index of a (possibly singular) -Gorenstein del Pezzo surface is the greatest rational number such that , where is a primitive Cartier divisor. This paper describes the set of values…

3-FOLD LOG FLIPS

- Mathematics
- 1993

We prove that 3-fold log flips exist. We deduce the existence of log canonical and -factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and log…

Log Del Pezzo surfaces III. Math. USSR Izvestia

- Log Del Pezzo surfaces III. Math. USSR Izvestia
- 1990

Classification of Del Pezzo surfaces with log terminal singularities of index ≤ 2 and involutions on K3 surfaces

- Soviet Math. Dokl. 39
- 1989

Ann of Math

- Ann of Math
- 1988

Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces

- Mathematics
- 1988

In: Birational geometry of algebraic varieties: open problems. 23 Taneguchi intnl symposium

- In: Birational geometry of algebraic varieties: open problems. 23 Taneguchi intnl symposium
- 1988