P-Resolutions of Cyclic Quotients from the Toric Viewpoint

Altmann, Klaus

doi:10.1007/978-3-0348-8770-0_12

Cited by 6 publications

(17 citation statements)

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“…. , u j + α h dv gives a P-resolution of Y ; the corresponding k ∈ K e−2 (Y ) must satisfy d = a h − k h , proving (2).…”

Section: And Y Is a Complete Intersection In Xmentioning

confidence: 93%

“…The general fiber of π 1 3 has the singularities (2, 2) and (2, 1), the second of which blows down to the smooth (1). Finally, the general fiber of π 2 3 has the singularities (1, 2) and (2,2), the first of which blows down to the smooth (1).…”

Section: Adjacencies and The General Fibermentioning

confidence: 97%

“…The corresponding sequences of α i , 2 ≤ i ≤ e − 1 are, respectively, (1, 1, 1) and (1, 2, 1). The fan giving the P-resolution corresponding to (1, 2, 1) is generated by rays through (1, 0), 1 8 (3, 1), 1 8 (1, 3), and (0, 1), while the fan giving the P-resolution corresponding to (2,1,2) is generated by rays through (1, 0) and (0, 1), that is, it is simply the original cone σ .…”

Section: Theorem 12 ([2]mentioning

confidence: 99%

“…The Hilbert basis of the dual semigroup consists of [0, 8], [1,5], [2,2], [5,1], and [8,0]. The continued fraction expansion of 8/5 is [2,3,2] and the elements of K 3 (Y ) are the chains (1, 2, 1) and (2,1,2). The corresponding sequences of α i , 2 ≤ i ≤ e − 1 are, respectively, (1, 1, 1) and (1, 2, 1).…”

Section: Theorem 12 ([2]mentioning

confidence: 99%

“…In [2], Altmann shows how to construct a Presolution as a toric variety given an element k ∈ K e−2 (Y ). We outline this construction:…”

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confidence: 99%

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One-parameter toric deformations of cyclic quotient singularities

Ilten

2009

Journal of Pure and Applied Algebra

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a b s t r a c tIn the case of two-dimensional cyclic quotient singularities, we classify all oneparameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions. IntroductionThe deformation theory of two-dimensional cyclic quotient singularities is well understood. Kollár and Shepherd-Barron showed a correspondence between certain partial resolutions (P-resolutions) and reduced versal base components in [8], and Arndt managed to write down equations for the versal deformation in [3]. Furthermore, Christophersen and Stevens were able to give much nicer equations for each reduced component in [5,9], respectively.Taking a slightly different viewpoint, we use the fact that two-dimensional cyclic quotient singularities correspond to two-dimensional affine varieties and consider one-parameter toric deformations introduced by Altmann in [1]. These deformations can be described simply in terms of Minkowski decompositions of line segments, yet contain much of the information present in the versal deformation. For a given singularity, we completely classify all such deformations. Furthermore, we show how to induce each deformation from a versal family, show to exactly which reduced versal base space components each deformation maps, calculate the singularities occurring in the general fiber, and construct corresponding partial simultaneous resolutions.In Section 1, we cover some preliminaries and introduce notation. Section 2 introduces toric deformations and classifies all possible one-parameter toric deformations for a given singularity. In Section 3, we construct maps from a versal family inducing these one-parameter toric deformations and identify all versal components to which each such deformation maps. In Section 4, we calculate the singularities occurring in the general fiber of a toric deformation. Finally, in Section 5, we show for each P-resolution how to construct simultaneous resolutions of each toric one-parameter deformation which maps to the corresponding versal base component. Cyclic quotients, P-resolutions, and chains representing zeroIn the following, we recall the notions of cyclic quotients, P-resolutions, continued fractions, and chains representing zero, as well as fixing notation. References are [7] for toric varieties, [8] for P-resolutions, and [4] for continued fractions and chains representing zero. Our notation is similar to that of [9,4].

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“…. , u j + α h dv gives a P-resolution of Y ; the corresponding k ∈ K e−2 (Y ) must satisfy d = a h − k h , proving (2).…”

Section: And Y Is a Complete Intersection In Xmentioning

confidence: 93%

Section: Adjacencies and The General Fibermentioning

confidence: 97%