Topology of Milnor fibers of minimally elliptic singularities

McEwan, Lee J.

doi:10.1090/S0002-9939-1993-1172960-1

Topology of Milnor fibers of minimally elliptic singularities
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Proc. Amer. Math. Soc. 118 (1993), 1017-1027 Request permission

Abstract:

Fundamental groups of Milnor fibers of some minimally elliptic singularities are computed via globalizations of smoothings. A conjecture of Looijenga-Wahl is verified for triangle singularities, which states that embedding dimension at most seven implies that any smoothing has simply-connected Milnor fiber.

References

Gert-Martin Greuel and Joseph Steenbrink, On the topology of smoothable singularities, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 535–545. MR 713090
K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR 0184257, DOI 10.2307/1970500
Eduard Looijenga, The smoothing components of a triangle singularity. I, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 173–183. MR 713246
E. Looijenga, The smoothing components of a triangle singularity. II, Math. Ann. 269 (1984), no. 3, 357–387. MR 761312, DOI 10.1007/BF01450700
Eduard Looijenga, Rational surfaces with an anticanonical cycle, Ann. of Math. (2) 114 (1981), no. 2, 267–322. MR 632841, DOI 10.2307/1971295
Eduard Looijenga and Jonathan Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986), no. 3, 261–291. MR 842425, DOI 10.1016/0040-9383(86)90044-3
Lee J. McEwan, Families of rational surfaces preserving a cusp singularity, Trans. Amer. Math. Soc. 321 (1990), no. 2, 691–716. MR 968419, DOI 10.1090/S0002-9947-1990-0968419-6
Rick Miranda and Ulf Persson, Configurations of $\textrm {I}_n$ fibers on elliptic $K3$ surfaces, Math. Z. 201 (1989), no. 3, 339–361. MR 999732, DOI 10.1007/BF01214900
Henry C. Pinkham, Smoothings of the $D_{pqr}$ singularities, $p+q+r=22$, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 373–377. MR 713261, DOI 10.1090/pspum/040.2/713261
Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI 10.2969/jmsj/02410020

On singular

surfaces