Tropical resolutions of configuration hypersurfaces

Referee: [Identification of normalized Nash blow-up with Bloch incidence variety] The central identification that the normalization of the Nash blow-up equals Bloch's incidence variety (for irreducible configuration hypersurfaces) is load-bearing for the whole recipe; the manuscript must supply a complete proof that the natural map is birational, normalizes the singularities, and coincides with the incidence variety, including any necessary checks on the defining ideals.

Authors: We agree this identification is central and that the current sketch needs expansion into a complete proof. The manuscript defines the natural map via the Nash blow-up and torus orbit, establishes birationality over the dense open set, and identifies the target with Bloch's incidence variety by matching incidence conditions. In revision we will add a self-contained subsection proving: (i) the map is birational by showing it induces an isomorphism on function fields and is an isomorphism over the smooth locus; (ii) it normalizes singularities by verifying that the coordinate ring of the image is the integral closure; (iii) the defining ideals coincide by explicit comparison in the ambient projective space using the configuration polynomial. These checks will be carried out for the general irreducible case. revision: yes

Referee: [Tropical compactification and morphism construction] The claim that the bipermutohedral fan yields a smooth tropical compactification via Tevelev's theorem requires explicit verification that the torus subvariety is schön (all initial degenerations are smooth), that the compactification is proper and birational onto the normalized Nash blow-up, and that the morphism resolves the singularities with the expected exceptional set. This step is the least secure part of the argument.

Authors: We acknowledge that the schön property and birationality of the morphism need explicit verification. The manuscript constructs the fan from the bipermutohedral matroid fan of Ardila-Denham-Huh and invokes Tevelev's theorem, arguing smoothness of initial degenerations from the matroid structure. In revision we will insert a new subsection that: (i) verifies all initial ideals define smooth varieties by direct combinatorial computation using the matroid flats; (ii) proves the resulting tropical compactification is proper and birational onto the normalized Nash blow-up by analyzing the torus-equivariant morphism and the exceptional divisors; (iii) confirms the morphism resolves singularities with the expected exceptional set. These steps will make the argument fully explicit. revision: yes

Referee: [F-rationality and singularity type] The deduction of strong F-regularity from F-rationality of the biprojective cone, and the subsequent inference of rational singularities over C, must be checked for compatibility with the specific equations of configuration hypersurfaces; any implicit assumptions on the characteristic or the cone structure need to be stated clearly.

Authors: We will clarify the characteristic assumptions and compatibility. The manuscript certifies F-rationality of the biprojective cone using the explicit form of the configuration polynomials. Strong F-regularity then follows from the general fact that F-rational rings are strongly F-regular in this setting. The passage to rational singularities over C is via the standard reduction-mod-p argument. In revision we will add a remark stating the positive-characteristic hypothesis explicitly, verify that the cone equations arising from configuration polynomials satisfy the required F-rationality criteria, and confirm that no additional assumptions on the cone structure are needed beyond those already used. revision: yes