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Mathematics of Computation

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Segre class computation and practical applications

Authors: Corey Harris and Martin Helmer
Journal: Math. Comp. 89 (2020), 465-491
MSC (2010): Primary 14Qxx, 13Pxx, 13H15, 14C17, 14C20, 68W30, 65H10
Published electronically: May 24, 2019
MathSciNet review: 4011552
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Abstract: Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class ${s}(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

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Additional Information

Corey Harris
Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway

Martin Helmer
Affiliation: Department  of  Mathematical  Sciences,  University  of  Copenhagen,  Universitetsparken 5, DK-2100 Copenhagen, Denmark
Address at time of publication: Mathematical Sciences Institute, Hanna Neumann Building, Science Road, The Australian National University, Canberra ACT 2601 Australia
MR Author ID: 884722

Received by editor(s): July 13, 2018
Received by editor(s) in revised form: February 15, 2019
Published electronically: May 24, 2019
Additional Notes: The first author was partially supported by the Bergen Research Foundation project grant “Algebraic and topological cycles in tropical and complex geometry”.
The second author was partially supported by the Independent Research Fund of Denmark during the preparation of this work.
Article copyright: © Copyright 2019 American Mathematical Society