Chern classes for graph hypersurfaces and deletion-contraction relations

Paolo Aluffi

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. 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'.