Communications in Number Theory and Physics

Volume 15 (2021)

Number 3

Graph hypersurfaces with torus action and a conjecture of Aluffi

Pages: 455 – 488



Graham Denham (Department of Mathematics, University of Western Ontario, London, ON, Canada)

Delphine Pol (Department of Mathematics, Technische Universität Kaiserslautern, Germany)

Mathias Schulze (Department of Mathematics, Technische Universität Kaiserslautern, Germany)

Uli Walther (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


Generalizing the $\star$‑graphs of Müller–Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.


configuration, matroid, star graph, Euler characteristic, Grothendieck ring, torus action, Feynman, Kirchhoff, Symanzik

2010 Mathematics Subject Classification

Primary 05C31. Secondary 13D15, 14M12, 14N20, 14R20, 81Q30.

Graham Denham is supported by NSERC of Canada.

Delphine Pol is supported by a Humboldt Research Fellowship for Postdoctoral Researchers.

Uli Walther is supported in part by the National Science Foundation, and by a Simons Foundation Collaboration Grant for Mathematicians.

Received 28 May 2020

Accepted 15 January 2021

Published 15 July 2021