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Chern Classes of Splayed Intersections

Published online by Cambridge University Press:  20 November 2018

Paolo Aluffi  and
Eleonore Faber
Paolo Aluffi
Affiliation:
Mathematics Department, Florida State University, Tallahassee FL 32306, USA. e-mail: aluffi@math.fsu.edu
Eleonore Faber
Affiliation:
Department of Computer and Mathematical Sciences, University of Toronto at Scarborough, Toronto, ON M1A 1C4 Institut Mittag-Leffler, Auravägen 17, SE-182-60 Djursholm, Sweden. email: efaber@math.toronto.edu
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Abstract

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We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern–Schwartz–MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a strong splayedness assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern–Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Keywords

14C1714J17Splayed intersectionChern-Schwartz-MacPherson classChern-Fulton classsplayed blowupSegre class
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1201 - 1218
Copyright
Copyright © Canadian Mathematical Society 2015

References

[Alu94] Aluffi, P., MacPherson's and Fulton's Chern classes of hypersurfaces. Internat. Math. Res. Notices 1994, no. 11, 455–465.Google Scholar
[Alu99] Aluffi, P., Chern classes for singular hypersurfaces. Trans. Amer. Math. Soc. 351(1999), no. 10, 3989–4026.http://dx.doi.org/10.1090/S0002-9947-99-02256-4 Google Scholar
[Alu03] Aluffi, P., Computing characteristic classes of projective schemes. J. Symbolic Comput. 35(2003), no. 1, 3–19. http://www.math.fsu.edu/~aluffi/CSM/CSM.html Google Scholar
[Alul3] Aluffi, P., Euler characteristics of general linear sections and polynomial Chern classes. Rend. Circ. Mat. Palermo (2) 62(2013), no. 1, 3–26.http://dx.doi.Org/10.1007/s12215-013-0106-x Google Scholar
[AF13] Aluffi, P. and Faber, E., Splayed divisors and their Chern classes. J. Lond. Math. Soc. (2) 88(2013), no. 2, 563–579. http://dx.doi.Org/10.1112/jlms/jdtO32 Google Scholar
[EHOO] Eisenbud, D. and Harris, J., The geometry of schemes. Graduate Texts in Mathematics, 197, Springer-Verlag, New York, 2000.Google Scholar
[Fab 13] Faber, E., Towards transversality of singular varieties: splayed divisors. Publ. Res. Inst. Math. Sci. 49(2013), no. 3, 393–412. http://dx.doi.org/10.4171/PRIMS/109 Google Scholar
[Ful84] Fulton, W., Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2, Springer-Verlag, Berlin, 1984.Google Scholar
[Ken90] Kennedy, G., MacPherson's Chern classes of singular algebraic varieties. Comm. Algebra 18(1990), no. 9, 2821–2839.http://dx.doi.org/10.1080/00927879008824054 Google Scholar
[KT96] Kleiman, S. and Thorup, A., Mixed Buchsbaum-Rim multiplicities. Amer. J. Math. 118(1996), no. 3, 529–569. http://dx.doi.org/10.1353/ajm.1996.0026 Google Scholar
[Kwi94] Kwieciński, M., Sur le transformé de Nash et la construction du graphe de MacPherson. Thèse, Université de Provence, 1994.Google Scholar
[LiO9] Li, L., Wonderful compactification of an arrangement of subvarieties. Michigan Math. J. 58(2009), no. 2, 535–563.http://dx.doi.org/10.1307/mmjV1250169076 Google Scholar
[Mac74] MacPherson, R. D., Chern classes for singular algebraic varieties. Ann. of Math. (2) 100(1974),423–432.http://dx.doi.Org/10.2307/1971080 Google Scholar
[Sch] Schürmann, J., A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes. arxiv:math/O2O2175Google Scholar
[Sch65a] Schwartz, M.-H., Classes caractéristiques définies par une stratification d'une variété analytique complexe. I. C. R. Acad. Sci. Paris 260(1965), 3262–3264.Google Scholar
[Sch65b] Schwartz, M.-H., Classes caractéristiques définies par une stratification d'une variété analytique complexe. II. C. R. Acad. Sci. Paris 260(1965), 3535–3537.Google Scholar