# 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}
}
• V. Alexeev
• Published 12 June 1992
• Mathematics
• Duke Mathematical Journal
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…
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## References

SHOWING 1-10 OF 16 REFERENCES
Threefolds whose canonical bundles are not numerically effective.
• S. Mori
• Mathematics
Proceedings 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
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
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
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
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