We propose an intersection-theoretic method to reduce questions in genus 0 logarithmic Gromov–Witten theory to questions in the Gromov–Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative geometry of rational curves with maximal contact orders along a simple normal crossings divisor and to recent questions about its relationship to local curve counting. Three results are established. We produce counterexamples to the local-logarithmic conjectures of van Garrel–Graber–Ruddat and Tseng–You. We prove that a weak form of the conjecture holds for product geometries. Finally, we explicitly determine the difference between local and logarithmic theories, in terms of relative invariants for which efficient algorithms are known. The polyhedral geometry of the tropical moduli of maps plays an essential and intricate role in the analysis.

Abstract. In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T -singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov.

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For X a smooth projective variety and D = D1 + . . . + Dn a simple normal crossings divisor, we establish a precise cycle-level correspondence between the genus zero local Gromov-Witten theory of the bundle ⊕ n i=1 OX(−Di) and the maximal contact Gromov-Witten theory of the multi-root stack X D, r . The proof is an implementation of the rank reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.