## 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

Zbl 1315.14009

### Software:

Bertini; CharacteristicClasses; Macaulay2; PHCpack; CSM-A; SageMath
Full Text:

### 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] Bürgisser, 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.