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Euler characteristic of a complete intersection

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Book cover Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

In this paper we study the behaviour of the degree of the Fulton–Johnson class of a complete intersection under a blow–up with a smooth center under the assumption that the strict transform is again a complete intersection. Our formula is a generalization of the genus formula for singular curves in smooth surfaces.

Mathematics Subject Classification (2010) 14B05, 14E15.

Partially supported by MNiSW grant N N201 388834

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References

  1. P. Aluffi, Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc. 351 (1999), 3989–4026.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput. 35 (2003), no. 1, 3–19.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.-P. Brasselet, From Chern classes to Milnor classes. A history of characteristic classes for singular varieties. Advanced Studies in Pure Math. 29, (2000), pp. 31–52

    MathSciNet  Google Scholar 

  4. J-P. Brasselet, D. Lehmann, J. Seade, T. Suwa, Milnor classes of local complete intersections, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1351–1371.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Cynk, Defect of a nodal hypersurface. Manuscripta Math. 104 (2001), 325–331.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Cynk, Cohomologies of a double covering of a non-singular algebraic 3-fold. Math. Z. 240 (2002), 731–743.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Cynk, D. van Straten, Infinitesimal deformations of double covers of smooth algebraic varieties. Math. Nach. 279 (2006), 716–726.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Cynk, S. Rams, Defect via differential forms with logarithmic poles, preprint arXiv:0909.4655.

    Google Scholar 

  9. H. Esnault, E. Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser, 1992.

    Google Scholar 

  10. W. Fulton, Intersection theory, Springer-Verlag, 1984.

    Google Scholar 

  11. W. Fulton, K. Johnson, Canonical classes on singular varieties, Manuscripta Math. 32 (1980), no. 3–4, 381–389.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Hartshorne, Algebraic geometry. Springer 1977.

    Google Scholar 

  13. F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer 1978.

    Google Scholar 

  14. G. Kapustka, Primitive contractions of Calabi-Yau threefolds. II. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 259–271.

    MATH  MathSciNet  Google Scholar 

  15. J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 Princeton University Press, 1968.

    Google Scholar 

  16. A. Parusiński, A generalization of the Milnor number. Math. Ann. 281 (1988), no. 2, 247–254.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Parusiński, P. Pragacz, A formula for the Euler characteristic of singular hypersurfaces, J. Algebraic Geom. 4 (1995), 337–351.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  19. S. Rams, Defect and Hodge numbers of hypersurfaces. Adv. Geom. 8 (2008), 257–288.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348, Kraków, Poland

    Sławomir Cynk

  2. Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warszawa, Poland

    Sławomir Cynk

Authors
  1. Sławomir Cynk
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Correspondence to Sławomir Cynk .

Editor information

Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

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Cynk, S. (2011). Euler characteristic of a complete intersection. 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_6

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