\(\pi_1\) of Miranda moduli spaces of elliptic surfaces. (English) Zbl 1499.14062
The author gives an explicit presentation of the fundamental group of the moduli space of (Weierstrass models) of elliptic surfaces over the projective line constructed by R. Miranda [Math. Ann. 255, 379–394 (1981; Zbl 0438.14023)] by relating it to a discriminant complement in a linear system on a Hirzebruch surface. He also discusses possible variants and generalisations of his approach.
Reviewer: Sönke Rollenske (Marburg)
MSC:
| 14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
| 32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |
Keywords:
elliptic surfaces; moduli spaces; orbifold fundamental group; braid monodromy; discriminant knot groupCitations:
Zbl 0438.14023References:
| [1] | Dolgachev, I., Libgober, A.: On the fundamental group of the complement to a discriminant variety, in Algebraic geometry, Chicago: LNM 862. Springer, Berlin, 1981, 1-25 (1980) · Zbl 0475.14011 |
| [2] | Catanese, F.; Wajnryb, B., The fundamental group of generic polynomials, Topology, 30, 4, 641-651 (1991) · Zbl 0755.57001 · doi:10.1016/0040-9383(91)90044-5 |
| [3] | Fadell, E.; Van Buskirk, J., On the braid groups of \(E^2\) and \(S^2\), Bull. AMS, 67, 211-213 (1961) · Zbl 0117.41003 · doi:10.1090/S0002-9904-1961-10570-3 |
| [4] | Hefez, A.; Lazzeri, F., The intersection matrix of Brieskorn singularities, Invent. Math., 25, 143-157 (1974) · Zbl 0288.32010 · doi:10.1007/BF01390172 |
| [5] | Lönne, M., Bifurcation braid monodromy of elliptic surfaces, Topology, 45, 4, 785-806 (2006) · Zbl 1099.14030 · doi:10.1016/j.top.2006.03.001 |
| [6] | Lönne, M., Fundamental groups of projective discriminant complements, Duke Math. J., 150, 2, 357-405 (2009) · Zbl 1202.14041 · doi:10.1215/00127094-2009-055 |
| [7] | Lönne, M., Braid monodromy of some Brieskorn-Pham singularities, Internat. J. Math., 21, 1047-1070 (2010) · Zbl 1207.32025 · doi:10.1142/S0129167X10006379 |
| [8] | Lönne, M., On a discriminant knot group problem of Brieskorn, J. Singul., 18, 455-463 (2018) · Zbl 1404.32053 |
| [9] | Looijenga, E., Artin groups and the fundamental groups of some moduli spaces, J. Topol., 1, 187-216 (2008) · Zbl 1172.14022 · doi:10.1112/jtopol/jtm009 |
| [10] | Miranda, R., The moduli of Weierstrass fibrations over \(P^1\), Math. Ann., 255, 379-394 (1981) · Zbl 0438.14023 · doi:10.1007/BF01450711 |
| [11] | Nori, M.: Zariski’s conjecture and related problems. Ann. Sci. Ecole Norm. Sup. 16, 305-344 (1983) · Zbl 0527.14016 |
| [12] | Seiler, W., Global moduli for polarized elliptic surfaces, Compos. Math., 62, 187-213 (1987) · Zbl 0624.14024 |
| [13] | Zariski, O., On the Poincaré group of rational plane curves, Am. J. Math., 58, 607-619 (1936) · JFM 62.0758.02 · doi:10.2307/2370979 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
