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 Mosc. Math. J., 2012, Volume 12, Number 4, Pages 671–700 (Mi mmj475)

Chern classes of graph hypersurfaces and deletion-contraction relations

Paolo Aluffi

Mathematics Department, Florida State University, Tallahassee FL 32306, U.S.A.

Abstract: We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove that both these conditions hold when the edge is multiple in the graph. This leads to recursions for the Chern classes of graph hypersurfaces for graphs obtained by adding parallel edges to a given (regular) edge.
Analogous results for the case of Grothendieck classes of graph hypersurfaces were obtained in previous work, and both Grothendieck classes and Chern classes were used to define ‘algebro-geometric’ Feynman rules. The results in this paper provide further evidence that the polynomial Feynman rule defined in terms of the Chern–Schwartz–MacPherson class of a graph hypersurface reflects closely the combinatorics of the corresponding graph.
The key to the proof of the main result is a more general formula for the Chern–Schwartz–MacPherson class of a transversal intersection (see Section 3), which may be of independent interest. We also describe a more geometric approach, using the apparatus of ‘Verdier specialization’.

Key words and phrases: Chern classes, graph hypersurfaces, deletion-contraction, Feynman rules.

DOI: https://doi.org/10.17323/1609-4514-2012-12-4-671-700

Full text: http://www.mathjournals.org/.../2012-012-004-001.html
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Bibliographic databases:

MSC: 14C17, 81Q30, 81T18, 05C75
Received: July 1, 2011; in revised form January 9, 2012
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Citation: Paolo Aluffi, “Chern classes of graph hypersurfaces and deletion-contraction relations”, Mosc. Math. J., 12:4 (2012), 671–700

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Aluffi P., Faber E., “Splayed Divisors and their Chern Classes”, J. Lond. Math. Soc.-Second Ser., 88:2 (2013), 563–579
2. P. Aluffi, “How many hypersurfaces does it take to cut out a Segre class?”, J. Algebra, 471 (2017), 480–491
3. Kulkarni A., Maxedon G., Yeats K., “Some Results on An Algebro-Geometric Condition on Graphs”, J. Aust. Math. Soc., 104:2 (2018), 218–254
4. Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59
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