Pinkham-Demazure construction for two dimensional cyclic quotient singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

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Functions on quotient singularities

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1987.0116 ◽

1987 ◽

Vol 324 (1576) ◽

pp. 1-45 ◽

Cited By ~ 4

Keyword(s):

Invariant Theory ◽

Quotient Singularity ◽

Two Dimensional ◽

Comprehensive Review ◽

Invariant Functions ◽

Quotient Singularities ◽

Free Actions

There are two main results: a determination of the modality of a generic function on any given two-dimensional quotient singularity and a listing of all the zero-modal functions. To achieve this, a comprehensive review of the invariant theory for free actions on C 2 is needed. The problem is put in context by a general discussion of classification of invariant functions, and several different extensions of the main results are indicated.

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On the Milnor fibres of cyclic quotient singularities

Proceedings of the London Mathematical Society ◽

10.1112/plms/pdq007 ◽

2010 ◽

Vol 101 (2) ◽

pp. 554-588 ◽

Cited By ~ 6

Author(s):

András Némethi ◽

Patrick Popescu-Pampu

Keyword(s):

Quotient Singularities ◽

Cyclic Quotient Singularities

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Stable quintuples and terminal quotient singularities

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100068389 ◽

1990 ◽

Vol 107 (1) ◽

pp. 91-101 ◽

Cited By ~ 3

Author(s):

G. K. Sankaran

Keyword(s):

Algebraic Geometry ◽

Geometry Of Numbers ◽

Quotient Singularities ◽

Cyclic Quotient Singularities

We shall prove below part of a conjecture made by Shigefumi Mori, David Morrison and Ian Morrison in the course of their investigations into the properties of isolated terminal cyclic quotient singularities of prime Gorenstein index in dimension four [1]. The reader of the present paper need have no knowledge of algebraic geometry, because we quickly reduce the problem to one about the geometry of numbers that can be solved by elementary calculations. The calculations are very lengthy and not quite routine, so what the reader does need is either patience, if he intends to check them, or faith, if he does not. We give only part of the calculations below. Full details may be obtained from the author.*

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