Abstract
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement about the interdependence of the two monodromy maps. Our study concludes with some implication with regard to the unfaithfulness of the geometric monodromy ([W]) and the – yet unexploited – knotted geometric monodromy, which takes the ambient space into account.
Mathematics Subject Classification (2010) Primary 32S50. Secondary 14D05, 32S55, 57Q45.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Emil Artin: Theory of Braids, Ann. Math. (2) 48 (1947), 101–126
Bessis, David: Zariski theorems and diagrams for braid groups, Invent. Math. 145 (2001), 487–507.
F. Catanese, B. Wajnryb: The fundamental group of generic polynomials, Topology 30 (1991), 641–651.
Dolgachev, Igor, Libgober, Anatoly: On the fundamental group of the complement to a discriminant variety, in Algebraic geometry, Chicago 1980, LNM 862, Springer, Berlin-New York, 1981, 1–25.
Hirose, Susumu: Surfaces in the complex projective plane and their mapping class groups, Algebr. Geom. Topol. 5 (2005), 577–613.
Lönne, Michael: Bifurcation braid monodromy of elliptic surfaces, Topology 45 (2006), 785–806.
Lönne, Michael: Versal braid monodromy, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 873–876.
Lönne, Michael: Fundamental groups of projective discriminant complements, Duke Math. J. 150 (2009), no. 2, 357–405.
Lönne, Michael: Braid Monodromy of some Brieskorn Pham Singularities, Internat. J. Math. 21 (2010), 1047–1070
Lönne, Michael: Geometric and algebraic significance of some Hurwitz stabilisers in Br n , Int. Electron. J. Algebra 8 (2010), 1–17.
E. Looijenga: The complement of the bifurcation variety of a simple singularity, Invent. Math. 23 (1974), 105–116.
Perron, B., Vannier, J. P.: Groupe de monodromie géométrique des singularités simples, J. Math. Ann. 306 (1996), 231–245.
Rudolph, Lee: Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), 1–37.
Wajnryb, Bronislaw: Artin groups and geometric monodromy, Invent. Math. 138 (1999), 563–571.
Zariski, Oscar: On the Poincaré group of rational plane curves, Am. J. Math. 58 (1936), 607–619.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lönne, M. (2011). Bifurcation braid monodromy of plane curves. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-20300-8_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20299-5
Online ISBN: 978-3-642-20300-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)