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A direct algorithm to compute the topological Euler characteristic and Chern-Schwartz-MacPherson class of projective complete intersection varieties. (English) Zbl 1378.14061

Summary: Let \(V\) be a possibly singular scheme-theoretic complete intersection subscheme of \(\mathbb{P}^n\) over an algebraically closed field of characteristic zero. Using a recent result of J. Fullwood [J. Singul. 8, 1–10 (2014; Zbl 1315.14009)] we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of \(V\). This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern-Schwartz-MacPherson class and Euler characteristic for certain types of complete intersection subschemes of \(\mathbb{P}^n\).

MSC:

14Q15 Computational aspects of higher-dimensional varieties
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
14M10 Complete intersections
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 1315.14009
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References:

[1] Aluffi, Paolo, Singular schemes of hypersurfaces, Duke Math. J., 80, 2, 325-352, (1995) · Zbl 0876.14028
[2] Aluffi, Paolo, Computing characteristic classes of projective schemes, J. Symbolic Comput., 35, 1, 3-19, (2003) · Zbl 1074.14502
[3] Aluffi, Paolo, Characteristic classes of singular varieties, (Top. in Cohomo. Studies of Alg. Var., (2005), Springer), 1-32 · Zbl 1292.14007
[4] Aluffi, Paolo, Euler characteristics of general linear sections and polynomial Chern classes, Rend. Circ. Mat. Palermo, 1-24, (2013) · Zbl 1312.14018
[5] Aluffi, Paolo; Esole, Mboyo, Chern class identities from tadpole matching in type IIB and F-theory, J. High Energy Phys., 3, (2009) · Zbl 1270.81145
[6] Aluffi, Paolo; Esole, Mboyo, New orientifold weak coupling limits in F-theory, J. High Energy Phys., 2010, 2, 1-53, (2010) · Zbl 1270.81145
[7] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, Charles W. Wampler, Bertini: software for numerical algebraic geometry. · Zbl 1143.65344
[8] B├╝rgisser, Peter; Cucker, Felipe; Lotz, Martin, Counting complexity classes for numeric computations. III: complex projective sets, Found. Comput. Math., 5, 4, 351-387, (2005) · Zbl 1100.68032
[9] Collinucci, Andres; Denef, Frederik; Esole, Mboyo, D-brane deconstructions in IIB orientifolds, J. High Energy Phys., 2009, 02, (2009) · Zbl 1245.81156
[10] DeMillo, Richard A.; Lipton, Richard J., A probabilistic remark on algebraic program testing, Inform. Process. Lett., 7, 4, 193-195, (1978) · Zbl 0397.68011
[11] Eklund, David; Jost, Christine; Peterson, Chris, A method to compute Segre classes of subschemes of projective space, J. Algebra Appl., 12, 02, 1250142, (2013) · Zbl 1274.13044
[12] Fullwood, James, On Milnor classes via invariants of singular subschemes, J. Singul., 8, 1-10, (2014) · Zbl 1315.14009
[13] Fulton, William, Intersection theory, (1998), Springer · Zbl 0885.14002
[14] Grayson, Daniel R.; Stillman, Michael E., Macaulay2, a software system for research in algebraic geometry, (2013)
[15] Harris, Joe, Algebraic geometry: A first course, vol. 133, (1992), Springer · Zbl 0779.14001
[16] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, (1977), Springer New York · Zbl 0367.14001
[17] Heintz, Joos, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci., 24, 3, 239-277, (1983) · Zbl 0546.03017
[18] Helmer, Martin, Algorithms to compute the topological Euler characteristic, Chern-Schwartz-macpherson class and Segre class of projective varieties, J. Symbolic Comput., 73, 120-138, (2016) · Zbl 1349.14028
[19] Huh, June, The maximum likelihood degree of a very affine variety, Compos. Math., 1-22, (2012)
[20] Jost, Christine, An algorithm for computing the topological Euler characteristic of complex projective varieties, (2013)
[21] MacPherson, Robert D., Chern classes for singular algebraic varieties, Ann. Math., 100, 2, 423-432, (1974) · Zbl 0311.14001
[22] Rayner, F. J., An algebraically closed field, Glasg. Math. J., 9, 146-151, (1968) · Zbl 0159.05302
[23] Schwartz, Jacob T., Fast probabilistic algorithms for verification of polynomial identities, J. ACM, 27, 4, 701-717, (1980) · Zbl 0452.68050
[24] Sommese, A. J.; Wampler, C. W., The numerical solution of systems of polynomials arising in engineering and science, (2005), World Scientific · Zbl 1091.65049
[25] Stein, W. A., Sage mathematics software (version 5.11), (2013), The Sage Development Team
[26] Suwa, Tatsuo, Classes de Chern des intersections completes locales, C. R. Acad. Sci. Ser. I: Math., 324, 1, 67-70, (1997) · Zbl 0957.14034
[27] Verschelde, Jan, Algorithm 795: phcpack: a general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software, 25, 2, 251-276, (1999) · Zbl 0961.65047
[28] Zippel, Richard, Probabilistic algorithms for sparse polynomials, (1979), Springer · Zbl 0418.68040
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