Abstract
Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note, we consider the problem of computing a particular characteristic class, the Chern–Schwartz–MacPherson class, of a complete simplicial toric variety \(X_{\Sigma }\) defined by a fan \({\Sigma }\) from the combinatorial data contained in the fan \(\Sigma \). Specifically, we give an effective combinatorial algorithm to compute the Chern–Schwartz–MacPherson class of \(X_{\Sigma }\), in the Chow ring (or rational Chow ring) of \(X_{\Sigma }\). This method is formulated by combining, and when necessary modifying, several known results from the literature and is implemented in Macaulay2 for test purposes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aluffi, P.: Computing characteristic classes of projective schemes. J. Symb. Comput. 35(1), 3–19 (2003)
Barthel, G., Brasselet, J.-P., Fieseler, K.-H.: Classes de Chern de variétés toriques singulières. CR Acad. Sci. Paris Sér. I Math. 315(2), 187–192 (1992)
Brasselet, J.-P., Schwartz, M.-H.: Sur les classes de Chern d’un ensemble analytique complexe. Astérisque 82(83), 93–147 (1981)
Cox, D.A., John, B., Schenck, H.K.: Toric varieties. Am. Math. Soc. 124, 575 (2011)
Fulton, W.: Intersection Theory, 2nd edn. Springer, Berlin (1998)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Biometrika 66(2), 339–344 (2013)
Helmer, M.: An algorithm to compute the topological Euler characteristic, the Chern–Schwartz–Macpherson class and the Segre class of subschemes of some smooth complete toric varieties. arXiv:1508.03785 (2015)
Helmer, M.: Algorithms to compute the topological Euler characteristic, Chern–Schwartz–Macpherson class and Segre class of projective varieties. J. Symb. Comput. 73, 120–138 (2015)
Helmer, M.: A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–Macpherson class of projective complete intersection varieties. Submitted to a Special Issue of the Journal of Theoretical Computer Science for SNC-2014. Available on the, arXiv.org/abs/1410.4113 (2015)
Jost, C.: An algorithm for computing the topological Euler characteristic of complex projective varieties. arXiv:1301.4128 (2013)
Robert, D.: MacPherson. Chern classes for singular algebraic varieties. Ann. Math. 100(2), 423–432 (1974)
Schürmann, J., Yokura, S.: A Survey of Characteristic Classes of Singular Spaces, pp. 865–952. World Scientific, Singapore (2007)
Schwartz, M.-H.: Classes caractéristiques définies par une stratification d’une variété analytique complexe. Comptes Rendus de l’Académie des Sciences Paris 260, 3262–3264 (1965)
Stein, W.A et al.: Sage Mathematics Software (Version 5.11). The Sage Development Team, http://www.sagemath.org (2013)
The LinBox Group.: LinBox–Exact Linear Algebra Over the Integers and Finite Rings, Version 1.1.6 (2008)
The PARI Group, Bordeaux.: PARI/GP version 2.7.0. Available from http://pari.math.u-bordeaux.fr/ (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Helmer, M. (2017). Computing the Chern–Schwartz–MacPherson Class of Complete Simplical Toric Varieties. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-56932-1_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56930-7
Online ISBN: 978-3-319-56932-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)