Cohomology of graph hypersurfaces associated to certain Feynman graphs

To any Feynman graph (with $2n$ edges) we can associate ahyper-surface $X\subset\PP^{2n-1}$. We study the cohomologyof the middle degree $H^{2n-2}(X)$ of such graph hypersurface.Bloch et al. (Commun. Math. Phys. 267, 2006)have computed this cohomology for the first series of examples, thewheel with spokes $WS_n$, $n\geq 3$. Using the same technique, weintroduce the generalized zigzag graphs and prove that$W_5(H^{2n-2}(X))=\QQ(-2)$ for all of them (with $W_{\ast}$ theweight filtration). We also can compute $\#X(\FF_q)\equiv1+q+2q^2\,{\rm mod}\, q^3$ for the number of rational points of suchhypersurface. At the end, we study the behavior of graphhypersurfaces under the gluing of graphs.

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Published 1 January 2010