Tropical resolutions of configuration hypersurfaces

TL;DR

This paper develops a two-step method to resolve singularities of configuration hypersurfaces, combining normalization, tropical compactification, and combinatorics, with implications for Feynman integrals.

Contribution

It introduces an explicit tropical compactification approach for resolving singularities of configuration hypersurfaces using bipermutohedral matroid combinatorics.

Findings

Normalized Nash blow-up has strongly F-regular singularities in positive characteristic.

Constructed explicit tropical compactification for configuration hypersurfaces.

Established rational singularities over the complex numbers for the normalized Nash blow-up.

Abstract

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly $F$-regular singularities in positive characteristic. We deduce this by certifying $F$-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.