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Vanishing homology of projective hypersurfaces with 1-dimensional singularities

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Abstract

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus \({\Sigma }\), and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of \({\Sigma }\) and on the effect of the monodromy of the local system over its complement.

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Fig. 1

Notes

  1. See [2] for details.

  2. The polar locus of a map (hf) is defined as \(\overline{\mathrm{Sing}\,(h,f) {\setminus }(\mathrm{Sing}\,h \cup \mathrm{Sing}\,f)}\).

  3. A related result was obtained in [12]. Like in case of [12], the proof actually works for any singular locus \(\mathrm{Sing}\,V\) and any general pencil.

  4. We recall from Sect. 5.2 that the notation \(S_q\) depends on whether the point q is considered in Q or in \(Q_i\), namely it takes either the local branches of \({\Sigma }\) at q, or the local branches of at q, accordingly.

  5. Note that no multiplicities but only transversal types are involved in the rank formula.

  6. Dimca states such a result [5, Theorem 4.1] referring to [4, p. 144] for Kato’s proof in cohomology [10].

References

  1. Damon, J., Pike, B.: Solvable groups, free divisors and nonisolated matrix singularities II: vanishing topology. Geom. Topol. 18(2), 911–962 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dimca, A.: On the homology and cohomology of complete intersections with isolated singularities. Compositio Math. 58(3), 321–339 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Dimca, A.: On the Milnor fibrations of weighted homogeneous polynomials. Compositio Math. 76(1–2), 19–47 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Dimca, A.: Singularities and Topology of Hypersurfaces. Universitext. Springer, New York (1992)

    Book  MATH  Google Scholar 

  5. Dimca, A.: Singularities and their deformations: how they change the shape and view of objects. In: Elkadi, M., Mourrain, B., Piene, R. (eds.) Algebraic Geometry and Geometric Modeling. Mathematics and Visualization, pp. 87–101. Springer, Berlin (2006)

    Chapter  Google Scholar 

  6. Frühbis-Krüger, A., Zach, M.: On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities (2015). arXiv:1501.01915

  7. Hulek, K., Kloosterman, R.: Calculating the Mordell–Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces. Ann. Inst. Fourier (Grenoble) 61(3), 1133–1179 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Iomdin, I.N.: Complex surfaces with a one-dimensional set of singularities. Siberian Math. J. 15(5), 748–762 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. de Jong, T.: Some classes of line singularities. Math. Z. 198(4), 493–517 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kato, M.: Topology of \(k\)-regular spaces and algebraic sets. In: Manifolds–Tokyo 1973, pp. 153–159. University of Tokyo Press, Tokyo (1975)

  11. Libgober, A.S., Wood, J.W.: On the topological structure of even-dimensional complete intersections. Trans. Amer. Math. Soc. 267(2), 637–660 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Parusiński, A., Pragacz, P.: Characteristic classes of hypersurfaces and characteristic cycles. J. Algebraic Geom. 10(1), 63–79 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Randell, R.: On the topology of non-isolated singularities. In: Cantrell, J.C. (ed.) Geometric Topology, pp. 445–473. Academic Press, New York (1979)

    Chapter  Google Scholar 

  14. Siersma, D.: Isolated line singularities. In: Peter, O. (ed.) Singularities, Part 2. Proceedings of Symposia in Pure Mathematics, vol. 40.2, pp. 485–496. American Mathematical Society, Providence (1983)

  15. Siersma, D.: Quasihomogeneous singularities with transversal type \(A_1\). In: Richard, R. (ed.) Singularities. Contemporary Mathematics, vol. 90, pp. 261–294. American Mathematical Society, Providence (1989)

    Chapter  Google Scholar 

  16. Siersma, D.: Variation mappings on singularities with a \(1\)-dimensional critical locus. Topology 30(3), 445–469 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Siersma, D.: The vanishing topology of non isolated singularities. In: Siersma, D., Wall, C.T.C., Zakalyukin, V. (eds.) New Developments in Singularity Theory. NATO Science Series II: Math. Phys. Chem., vol. 21, pp. 447–472. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  18. Siersma, D., Tibăr, M.: Betti bounds of polynomials. Mosc. Math. J. 11(3), 599–615 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Siersma, D., Tibăr, M.: Milnor fibre homology via deformation. In: Decker, W., Pfister, G., Schulze, M. (eds.) Singularities and Computer Algebra, pp. 306–322. Springer, Cham (2017)

  20. Tibăr, M.: The vanishing neighbourhood of non-isolated singularities. Israel J. Math. 157, 309–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zach, M.: Vanishing cycles of smoothable isolated Cohen–Macaulay codimension 2 singularities of type 2 (2016). arXiv:1607.07527v1

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Correspondence to Mihai Tibăr.

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The authors express their gratitude to CIRM in Luminy for supporting this research project through the “Recherche en Binôme” program.

Mihai Tibǎr received the support from Labex CEMPI (ANR-11-LABX-0007-01).

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Siersma, D., Tibăr, M. Vanishing homology of projective hypersurfaces with 1-dimensional singularities. European Journal of Mathematics 3, 565–586 (2017). https://doi.org/10.1007/s40879-017-0151-7

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  • DOI: https://doi.org/10.1007/s40879-017-0151-7

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