# Gromov-Witten theory with maximal contacts

## TL;DR

This paper introduces an intersection-theoretic approach to connect genus zero logarithmic Gromov-Witten theory with smooth pair Gromov-Witten theory, providing new insights and counterexamples in enumerative geometry involving maximal contact orders.

## Contribution

It develops a novel method to reduce complex logarithmic Gromov-Witten questions to simpler cases and explicitly characterizes the difference between local and logarithmic theories.

## Key findings

- Counterexamples to local/logarithmic conjectures of van Garrel-Graber-Ruddat and Tseng-You.
- A weak form of the conjecture holds for product geometries.
- Explicit determination of the difference between local and logarithmic theories.

## Abstract

We propose an intersection-theoretic method to reduce questions in genus zero 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.