Computing characteristic classes of projective schemes. (English) Zbl 1074.14502

Summary: We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, non-reduced) projective scheme. For example, the algorithm yields the topological Euler characteristic of the support of a projective scheme \(S\), given the homogeneous ideal of \(S\). The algorithm has been implemented in Macaulay2.


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14Q15 Computational aspects of higher-dimensional varieties
68W30 Symbolic computation and algebraic computation


Macaulay2; CSM-A
Full Text: DOI arXiv


[1] Aluffi, P., Macpherson’s and fulton’s Chern classes of hypersurfaces, Int. math. res. notices, 11, 455-465, (1994) · Zbl 0839.14035
[2] Aluffi, P., Characteristic classes of discriminants and enumerative geometry, Comm. algebra., 26, 3165-3193, (1998) · Zbl 0934.14039
[3] Aluffi, P., Chern classes for singular hypersurfaces, Trans. am. math. soc., 351, 3989-4026, (1999) · Zbl 0972.57015
[4] Aluffi, P., 2002. Inclusion-exclusion and Segre classes. FSU02-11, arXiv:math.AG/0203122 (will appear in Comm. Algebra, issue dedicated to Steven Kleiman)
[5] Aluffi, P.; Faber, C., Linear orbits of d-tuples of points in {\bfp}1, J. reine angew. math., 445, 205-220, (1993) · Zbl 0781.14036
[6] Brasselet, J.-P., From Chern classes to Milnor classes—a history of characteristic classes for singular varieties, (), 31-52 · Zbl 1036.32022
[7] Eisenbud, D.; Grayson, D.; Stillman, M.; Sturmfels, B., Computations in algebraic geometry with Macaulay 2, (2002), Springer Berlin · Zbl 0973.00017
[8] Fulton, W., Intersection theory, (1984), Springer Berlin · Zbl 0541.14005
[9] Kennedy, G., Macpherson’s Chern classes of singular algebraic varieties, Comm. algebra., 18, 2821-2839, (1990) · Zbl 0709.14016
[10] MacPherson, R.D., Chern classes for singular algebraic varieties, Ann. math. (2), 100, 423-432, (1974) · Zbl 0311.14001
[11] Micali, A., Sur LES algèbres universelles, Ann. inst. Fourier (Grenoble), 14, 33-87, (1964) · Zbl 0152.02602
[12] Parusiński, A., A generalization of the Milnor number, Math. ann., 281, 247-254, (1988) · Zbl 0617.32012
[13] Parusiński, A.; Pragacz, P., Characteristic classes of hypersurfaces and characteristic cycles, J. algebraic. geom., 10, 63-79, (2001) · Zbl 1072.14505
[14] Schürmann, J., 2002. A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes, arXiv:math.AG/0202175
[15] Vasconcelos, W.V., 1998. Computational Methods in Commutative Algebra and Algebraic Geometry (with chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. Springer, Berlin) · Zbl 0896.13021
[16] Walther, U., Algorithmic determination of the rational cohomology of complex varieties via differential forms, (), 185-206 · Zbl 1079.14507
[17] Yokura, S., On a verdier-type riemann – roch for chern – schwartz – macpherson class, Topology appl., 94, 315-327, (1999), Special issue in memory of B.J. Ball · Zbl 0928.14010
[18] Yokura, S., On characteristic classes of complete intersections, (), 349-369 · Zbl 0966.14004
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