Abstract
AbstractWe 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.
Highlights
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
If D is smooth, logarithmic Gromov–Witten theory coincides with the relative theory for all tangency orders
If the tangency with D is maximal, it coincides with the local theory by a result of van Garrel–Graber–Ruddat [31], following Takahashi and Gathmann [28, 16]
Summary
The moduli space K(P) is smooth and so Δ is a regular embedding. We obtain a virtual class on N(P|H) by pullback: k [N(P|H)]vir := Δ! [K(P|Hi)]. Strong form: There is an equality of homology classes on the Kontsevich space K0,k (X, β) of k-pointed stable maps to X, suppressing the relevant pushforwards, given by k [K0m,akx ( X |D, β)]vir = (−1)di+1ev★i Di · [K0,k (⊕ik=1OX (−Di), β)]vir. After pushforward to K0,0 (X, β), the naive theory coincides up to explicit multiplicity with the local theory of (X |D) as a homology class on K0,0 (X, β). With this observation at hand, we dispense with local Gromov–Witten theory and focus on the more general question of when the logarithmic and naive theories coincide
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