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

This article is cited in 4 scientific papers (total in 4 papers)

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
References: PDF file   HTML file

Bibliographic databases:

MSC: 14C17, 81Q30, 81T18, 05C75
Received: July 1, 2011; in revised form January 9, 2012
Language:

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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\by Paolo~Aluffi
\paper Chern classes of graph hypersurfaces and deletion-contraction relations
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 4
\pages 671--700
\mathnet{http://mi.mathnet.ru/mmj475}
\crossref{https://doi.org/10.17323/1609-4514-2012-12-4-671-700}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3076849}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000314341500001}


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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  crossref  mathscinet  zmath  isi  scopus
    2. P. Aluffi, “How many hypersurfaces does it take to cut out a Segre class?”, J. Algebra, 471 (2017), 480–491  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    4. Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59  crossref  mathscinet  zmath  isi  scopus
  • Moscow Mathematical Journal
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