Abstract
We describe the versal deformation of two-dimensional cyclic quotient singularities in terms of equations, following Arndt, Brohme and Hamm. For the reduced components the equations are determined by certain systems of dots in a triangle. The equations of the versal deformation itself are governed by a different combinatorial structure, involving rooted trees.
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References
[1] Arndt, Jürgen, Verselle Deformationen zyklischer Quotientensingularitäten, Diss. Hamburg, 1988.
[2] Behnke, Kurt and Riemenschneider, Oswald, Quotient Surface Singularities and Their Deformations, in: Singularity theory (Trieste, 1991), 1–54, World Sci. Publ., River Edge, NJ, 1995.
[3] Brohme, Stephan, Monodromieüberlagerung der versellen Deformation zyklischer Quotientensingularitäten Diss. Hamburg 2002. URN: urn:nbn:de:gbv:18-6733.
[4] Christophersen, Jan, On the components and discriminant of the versal base space of cyclic quotient singularities, in: Singularity theory and its applications, Part I (Coventry, 1988/1989), 81–92, Lecture Notes in Math., 1462, Springer, Berlin, 1991.
[5] Christophersen, Jan, The Combinatorics of the Versal Base Space of Cyclic Quo-tient Singularities, in: Obstruction spaces for rational singularities and deformations of cyclic quotients. Thesis, Oslo s.a.
[6] Hamm, Martin, Die verselle Deformation zyklischer Quotientensingularitäten: Gleichungen und torische Struktur. Diss. Hamburg 2008. URN: urn:nbn:de:gbv:18-37828.
[7] Kollár, J. and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities. Invent. math. 91 (1986), 299–338.
[8] Riemenschneider, Oswald, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.
[9] Riemenschneider, Oswald, Zweidimensionale Quotientensingularitäten: Gleichungen und Syzygien, Arch. Math., 37 (1981), 406–417.
[10] Stanley, Richard P., Enumerative combinatorics. Vol. 2., Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.
[11] Stevens, Jan, On the versal deformation of cyclic quotient singularities, in: Singularity theory and its applications, Part I (Coventry, 1988/1989), 302–319, Lecture Notes in Math., 1462, Springer, Berlin, 1991.
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© 2013 János Bolyai Mathematical Society and Springer-Verlag
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Stevens, J. (2013). The Versal Deformation of Cyclic Quotient Singularities. In: Némethi, A., Szilárd, á. (eds) Deformations of Surface Singularities. Bolyai Society Mathematical Studies, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39131-6_5
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DOI: https://doi.org/10.1007/978-3-642-39131-6_5
Publisher Name: Springer, Berlin, Heidelberg
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