Fano Newton polytope and Q-Gorenstein degeneration from theta functions and mutations

Prove that for any smooth log Calabi–Yau pair (X, D) and any bounded chamber of the scattering diagram associated to (X, D), the Newton polytope of the theta function 11(X, D) is a Fano polytope; furthermore, establish that X admits a Q‑Gorenstein degeneration to the toric variety determined by the spanning fan of this Newton polytope, and demonstrate that two (not necessarily Fano) varieties are Q‑deformation equivalent if and only if their theta functions 11 are mutation equivalent.

Background

Section 11 develops the relationship between wall crossing, mutations, and theta functions in the scattering diagram framework. Mutations act on Laurent polynomials and preserve classical periods, suggesting deep ties between algebraic mutations and geometric degenerations.

Proposition 11.8 shows that the mutation of a Fano polytope remains Fano, and Corollary 11.6 relates mutation equivalence of theta functions to equality of periods and certain enumerative invariants. Building on these, Conjecture 11.9 proposes a global picture: the Newton polytope of the theta function in any bounded chamber should itself be Fano and define a toric model to which X degenerates via a Q‑Gorenstein degeneration, with mutation equivalence of theta functions characterizing Q‑deformation equivalence.

The authors verify this conjectural picture for Hirzebruch surfaces in Section 13, but leave the general case open.