Disentanglements of Corank 2 Map-Germs: Two Examples

  • David MondEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)


We compute the homology of the multiple point spaces of stable perturbations of two germs \((\mathbb {C}^n,0){\ \rightarrow \ }(\mathbb {C}^{n+1},0)\) of corank 2, using a variety of techniques based on the image computing spectral sequence ICSS. We provide a reasonably detailed introduction to the ICSS, including some low-dimensional examples of its use. The paper is partly expository.


Disentanglement Multiple-point spaces 

1991 Mathematics Subject Classification

14B05 32S30 32S25 



The calculations in Sect. 3 were begun in collaboration with Isaac Bird, as part of his final year MMath project at Warwick. I am grateful to him for agreement to use them here, and for his enthusiasm on the project, which contributed a great deal to its further development. I also thank Mike Stillman for help finding the \(2\times 4\) matrices of Sect. 4.2, and Juan José Nuño for pointing out the results of Greuel and Steenbrink in [9].


  1. 1.
    Altıntaş, A., Mond, D.: Free resolutions for multiple point spaces. Geom. Dedicata 162, 177–190 (2013). MR 3009540Google Scholar
  2. 2.
    Cooper, T., Mond, D., Wik Atique, R.: Vanishing topology of codimension 1 multi-germs over \(\mathbb{R}\) and \(\mathbb{C}\). Compos. Math. 131(2), 121–160 (2002). MR 1898432 (2004c:32052)Google Scholar
  3. 3.
    de Jong, T.: The virtual number of \(D_\infty \) points. I. Topology 29(2), 175–184 (1990). MR 1056268 (91f:32043)Google Scholar
  4. 4.
    de Jong, T., van Straten, D.: Disentanglements. Singularity Theory and Its Applications, Part I (Coventry, 1988/1989). Lecture Notes in Mathematics, vol. 1462, pp. 199–211. Springer, Berlin (1991). MR 1129033 (93a:14003)Google Scholar
  5. 5.
    Fruehbis-Krueger, A., Zach, M.: On the vanishing topology of isolated Cohen-Macaulay codimension 2 singularities (2015). arXiv:1501.01915
  6. 6.
    Goryunov, V.V.: Semi-simplicial resolutions and homology of images and discriminants of mappings. Proc. Lond. Math. Soc. (3) 70(2), 363–385 (1995). MR 1309234 (95j:32050)Google Scholar
  7. 7.
    Goryunov, V.V., Mond, D.: Vanishing cohomology of singularities of mappings. Compos. Math. 89(1), 45–80 (1993). MR 1248891 (94k:32058)Google Scholar
  8. 8.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry.
  9. 9.
    Greuel, G.-M., Steenbrink, J.: On the topology of smoothable singularities. Singularities, Part 1 (Arcata, California, 1981). Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 535–545. American Mathematical Society, Providence (1983). MR 713090Google Scholar
  10. 10.
    Houston, K.: Local topology of images of finite complex analytic maps. Topology 36(5), 1077–1121 (1997). MR 1445555 (98g:32064)Google Scholar
  11. 11.
    Kleiman, S.L.: Multiple-point formulas. I. Iteration. Acta Math. 147(1–2), 13–49 (1981). MR 631086 (83j:14006)Google Scholar
  12. 12.
    Mather, J.N.: On Thom-Boardman singularities. Dynamical Systems (Proceedings of a Symposium Held at the University of Bahia, Salvador, 1971), pp. 233–248. Academic Press, New York (1973). MR 0353359 (50 #5843)Google Scholar
  13. 13.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)Google Scholar
  14. 14.
    Marar, W.L., Mond, D.: Multiple point schemes for corank \(1\) maps. J. Lond. Math. Soc. (2) 39(3), 553–567 (1989). MR 1002466 (91c:58010)Google Scholar
  15. 15.
    Marar, W.L., Mond, D.: Real map-germs with good perturbations. Topology 35(1), 157–165 (1996). MR 1367279Google Scholar
  16. 16.
    Marar, W.L., Nuño-Ballesteros, J.J.: A note on finite determinacy for corank 2 map germs from surfaces to 3-space. Math. Proc. Camb. Philos. Soc. 145(1), 153–163 (2008). MR 2431646Google Scholar
  17. 17.
    Marar, W.L., Nuño-Ballesteros, J.J., Peñafort-Sanchis, G.: Double point curves for corank 2 map germs from \(\mathbb{C}^2\) to \(\mathbb{C}^3\). Topol. Appl. 159(2), 526–536 (2012). MR 2868913Google Scholar
  18. 18.
    Mond, D.: Some remarks on the geometry and classification of germs of maps from surfaces to \(3\)-space. Topology 26(3), 361–383 (1987). MR 899055Google Scholar
  19. 19.
    Mond, D.: Some open problems in the theory of singularities of mappings. J. Singul. 12, 141–155 (2015). MR 3317146Google Scholar
  20. 20.
    Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. Algebraic Geometry and Complex Analysis (Pátzcuaro, 1987). Lecture Notes in Mathematics, vol. 1414, pp. 107–161. Springer, Berlin (1989). MR 1042359 (91e:32035)Google Scholar
  21. 21.
    Nuño-Ballesteros, J.J., Peñafort-Sanchis, G.: On multiple point schemes (2015). arXiv:1509.04990
  22. 22.
    Némethi, A.: Invariants of normal surface singularities. Real and Complex Singularities. Contemporary Mathematics, vol. 354, pp. 161–208. American Mathematical Society, Providence (2004). MR 2087811Google Scholar
  23. 23.
    Serre, J.-P.: Algèbre locale. Multiplicité. Lecture Notes in Mathematics, vol. 11. Springer, Berlin (1957)Google Scholar
  24. 24.
    Sharland, A.A.: Examples of finitely determined map-germs of corank 2 from \(n\)-space to \((n+1)\)-space. Int. J. Math. 25(5), 1450044, 17 (2014). MR 3215220Google Scholar
  25. 25.
    Siersma, D.: Vanishing cycles and special fibres. Singularity Theory and Its Applications, Part I (Coventry, 1988/1989). Lecture Notes in Mathematics, vol. 1462, pp. 292–301. Springer, Berlin (1991). MR 1129039 (92j:32129)Google Scholar

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Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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