One-parameter toric deformations of cyclic quotient singularities

Abstract

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].

Maximally mutable Laurent polynomials

“…Each edge of a Fano polygon P ⊂ N Q determines a point, which may be singular, in the toric surface X P . We first describe the effect of mutation on these singularities: see [7,20,27].…”

Deformations of Rational T-Varieties

“…Remark 2.13. In section 4 of [Ilt09], explicit equations were used to calculate the singularities in the general fiber for toric deformations of cyclic quotient singularities. Combining this with the description of affine toric deformations in the following section, the above corollary provides a way of doing this without using the equations.…”

“…In [Mav04] and [Mav05], A. Mavlyutov constructed certain deformations of complete weak Fano toric varieties via, respectively, regluing an open cover with automorphisms, and representing one toric variety as a complete intersection inside of a larger toric variety. Furthermore, in [Ilt09], the first author constructed toric Q-Gorenstein deformations for partial resolutions of toric surface singularities.…”

Deformations of smooth toric surfaces

“…For Y non-complete and non-smooth, the author employed exactly such a construction in[Ilt09] to describe certain simultaneous resolutions.…”