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1
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On equivalences of derived and singular categories

100%
Baranovsky V., Pecharich J.
Open Mathematics
|
2010
|
tom 8
|
nr 1
1-14
EN
Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $$ \mathbb{A}^1 $$, g:Y → $$ \mathbb{A}^1 $$. Assuming that there exists a complex of sheaves on X × $$ \mathbb{A}^1 $$ Y which induces an equivalence of D b(X) and D b(Y), we show that there is also an equivalence of the singular derived categories of the fibers f −1(0) and g −1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.
2
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A bound for the Milnor number of plane curve singularities

81%
Płoski A.
Open Mathematics
|
2014
|
tom 12
|
nr 5
688-693
EN
Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].
3
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The jump of the Milnor number in the X 9 singularity class

52%
Brzostowski S., Krasiński T.
Open Mathematics
|
2014
|
tom 12
|
nr 3
429-435
EN
The jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.
4
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Three-dimensional terminal toric flips

52%
Fujino O., Sato H., Takano Y., Uehara H.
Open Mathematics
|
2009
|
tom 7
|
nr 1
46-53
EN
We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips.
5
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Weakly-exceptional quotient singularities

42%
Sakovics D.
Open Mathematics
|
2012
|
tom 10
|
nr 3
885-902
EN
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D n, E 6, E 7, E 8. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.
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