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Computing the Chern-Schwartz-MacPherson Class of complete simplical toric varieties. (English) Zbl 1383.14017
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 207-217 (2017).
Summary: 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.
For the entire collection see [Zbl 1379.13001].
MSC:
14Q15 Computational aspects of higher-dimensional varieties
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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