Stable quintuples and terminal quotient singularities

1990 ◽  
Vol 107 (1) ◽  
pp. 91-101 ◽  
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.*

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
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.

scholarly journals PRIMES REPRESENTED BY INCOMPLETE NORM FORMS

2020 ◽  
Vol 8 ◽  
Author(s):  
JAMES MAYNARD
Keyword(s):  
Algebraic Geometry ◽  
Irreducible Polynomial ◽  
Type I ◽  
Type Ii ◽  
Geometry Of Numbers ◽  
Asymptotic Number ◽  
Special Case

Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$ . We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$ . In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$ , we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$ . Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^{2}+Y^{4}$ and of Heath-Brown on $X^{3}+2Y^{3}$ . Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.

scholarly journals On the Milnor fibres of cyclic quotient singularities

2010 ◽  
Vol 101 (2) ◽  
pp. 554-588 ◽  
Author(s):  
András Némethi ◽  
Patrick Popescu-Pampu
Keyword(s):  
Quotient Singularities ◽  
Cyclic Quotient Singularities
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