Geodesic
Chern Classes of Splayed Intersections
Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1201-1218

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2015-010-7
Mots-clés : 14C17, 14J17, Splayed intersection, Chern-Schwartz-MacPherson class, Chern-Fulton class, splayed blowup, Segre class 
Aluffi, Paolo; Faber, Eleonore. Chern Classes of Splayed Intersections. Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1201-1218. doi: 10.4153/CJM-2015-010-7
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[Alu94] [Alu94] Aluffi, P., MacPherson's and Fulton's Chern classes of hypersurfaces. Internat. Math. Res. Notices 1994, no. 11, 455–465. Google Scholar

[Alu99] [Alu99] Aluffi, P., Chern classes for singular hypersurfaces. Trans. Amer. Math. Soc. 351(1999), no. 10, 3989–4026. Google Scholar | DOI

[Alu03] [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] [Alul3] Aluffi, P., Euler characteristics of general linear sections and polynomial Chern classes. Rend. Circ. Mat. Palermo (2) 62(2013), no. 1, 3–26. Google Scholar | DOI

[AF13] [AF13] Aluffi, P. and Faber, E., Splayed divisors and their Chern classes. J. Lond. Math. Soc. (2) 88(2013), no. 2, 563–579. Google Scholar | DOI

[EHOO] [EHOO] Eisenbud, D. and Harris, J., The geometry of schemes. Graduate Texts in Mathematics, 197, Springer-Verlag, New York, 2000. Google Scholar

[Fab 13] [Fab 13] Faber, E., Towards transversality of singular varieties: splayed divisors. Publ. Res. Inst. Math. Sci. 49(2013), no. 3, 393–412. Google Scholar | DOI

[Ful84] [Ful84] Fulton, W., Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2, Springer-Verlag, Berlin, 1984. Google Scholar

[Ken90] [Ken90] Kennedy, G., MacPherson's Chern classes of singular algebraic varieties. Comm. Algebra 18(1990), no. 9, 2821–2839. Google Scholar | DOI

[KT96] [KT96] Kleiman, S. and Thorup, A., Mixed Buchsbaum-Rim multiplicities. Amer. J. Math. 118(1996), no. 3, 529–569. Google Scholar | DOI

[Kwi94] [Kwi94] Kwieciński, M., Sur le transformé de Nash et la construction du graphe de MacPherson. Thèse, Université de Provence, 1994. Google Scholar

[LiO9] [LiO9] Li, L., Wonderful compactification of an arrangement of subvarieties. Michigan Math. J. 58(2009), no. 2, 535–563. Google Scholar | DOI

[Mac74] [Mac74] MacPherson, R. D., Chern classes for singular algebraic varieties. Ann. of Math. (2) 100(1974),423–432. Google Scholar | DOI

[Sch] [Sch] Schürmann, J., A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes. arxiv:math/O2O2175 Google Scholar

[Sch65a] [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] [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

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