Families of rational surfaces preserving a cusp singularity

McEwan, Lee J.

doi:10.1090/S0002-9947-1990-0968419-6

Families of rational surfaces preserving a cusp singularity
HTML articles powered by AMS MathViewer

by Lee J. McEwan

Trans. Amer. Math. Soc. 321 (1990), 691-716

DOI: https://doi.org/10.1090/S0002-9947-1990-0968419-6

PDF | Request permission

Abstract:

Families of rational surfaces containing resolutions of cusp singularities are explicitly constructed. It is proved that the families constructed are universal deformations at each point. Two different monodromy formulas are established; one of these is shown to be connected to automorphisms of Inoue-Hirzebruch surfaces. Some evidence (but no proof) is offered for the conjecture that finite base changes of the families we construct are the versal-deformation spaces for singular Inoue-Hirzebruch surfaces.

References

Robert Friedman and Rick Miranda, Smoothing cusp singularities of small length, Math. Ann. 263 (1983), no. 2, 185–212. MR 698002, DOI 10.1007/BF01456880
Robert Friedman and Henry Pinkham, Smoothings of cusp singularities via triangle singularities, Compositio Math. 53 (1984), no. 3, 303–324. With an appendix by Pinkham. MR 768826
Friedrich E. P. Hirzebruch, Hilbert modular surfaces, Série des Conférences de l’Union Mathématique Internationale, No. 4, Université de Genève, Secrétariat de l’Enseignement Mathématique, Geneva, 1973. MR 0389921
M. Inoue, New surfaces with no meromorphic functions. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 91–106. MR 0442297
Henry B. Laufer, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131–164. MR 333238, DOI 10.1007/BF01350842

Deformations of resolutions of two-dimensional singularities

Eduard Looijenga, Rational surfaces with an anticanonical cycle, Ann. of Math. (2) 114 (1981), no. 2, 267–322. MR 632841, DOI 10.2307/1971295
Ulf Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189, xv+144. MR 466149, DOI 10.1090/memo/0189
H. C. Pinkham, Automorphisms of cusps and Inoue-Hirzebruch surfaces, Compositio Math. 52 (1984), no. 3, 299–313. MR 756724
Jonathan Wahl, Smoothings of normal surface singularities, Topology 20 (1981), no. 3, 219–246. MR 608599, DOI 10.1016/0040-9383(81)90001-X