Advertisement

Inventiones mathematicae

, Volume 216, Issue 1, pp 153–213 | Cite as

Monodromy and vanishing cycles in toric surfaces

  • Nick SalterEmail author
Article
  • 279 Downloads

Abstract

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group.

Notes

Acknowledgements

The author would like to extend his warmest thanks to R. Crétois and L. Lang for helpful discussions of their work. He would also like to acknowledge C. McMullen for some insightful comments on a preliminary draft, and M. Nichols for a productive conversation. A special thanks is due to an anonymous referee for a very careful reading of the preprint and for many useful suggestions, both mathematical and expository.

References

  1. 1.
    Chillingworth, D.: Winding numbers on surfaces. I. Math. Ann. 196, 218–249 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chillingworth, D.: Winding numbers on surfaces. II. Math. Ann. 199, 131–153 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Crétois, R., Lang, L.: The Vanishing Cycles of Curves in Toric Surfaces I. Preprint. arXiv:1701.00608v2.pdf (2017)
  4. 4.
    Crétois, R., Lang, L.: The Vanishing Cycles of Curves in Toric Surfaces II. Preprint. arXiv:1706.07252.pdf (2017)
  5. 5.
    Dieudonné, J.: Sur les groupes classiques. Hermann, Paris (1973). Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, VI, Actualités Scientifiques et Industrielles, No. 1040Google Scholar
  6. 6.
    Donaldson, S.: Polynomials, vanishing cycles and Floer homology. In: Mathematics: Frontiers and Perspectives, pp. 55–64. American Mathematical Society, Providence (2000)Google Scholar
  7. 7.
    Farb, B., Margalit, D.: A Primer on Mapping Class Groups, Volume 49 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ (2012)Google Scholar
  8. 8.
    Grove, L.C.: Classical Groups and Geometric Algebra, Volume 39 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2002)Google Scholar
  9. 9.
    Humphries, S., Johnson, D.: A generalization of winding number functions on surfaces. Proc. Lond. Math. Soc. (3) 58(2), 366–386 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johnson, D.: An abelian quotient of the mapping class group \({\cal{I}}_{g}\). Math. Ann. 249(3), 225–242 (1980)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Johnson, D.: Spin structures and quadratic forms on surfaces. J. Lond. Math. Soc. (2) 22(2), 365–373 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Johnson, D.: A survey of the Torelli group. In: Low-Dimensional Topology (San Francisco, Calif., 1981), Volume 20 of Contemporary Mathematics, pp. 165–179. American Mathematical Society, Providence (1983)Google Scholar
  13. 13.
    Johnson, D.: The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves. Topology 24(2), 113–126 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Johnson, D.: The structure of the Torelli group. III. The abelianization of \(\mathscr {T}\). Topology 24(2), 127–144 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matsumoto, M.: A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities. Math. Ann. 316(3), 401–418 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Putman, A.: A note on the connectivity of certain complexes associated to surfaces. Enseign. Math. (2) 54(3–4), 287–301 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Salter, N.: On the Monodromy Group of the Family of Smooth Plane Curves. Preprint. arXiv:1610.04920.pdf (2016)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations