Gromov-Witten theory with maximal contacts

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.

34 pages, 1 figure. Final version to appear in Forum of Mathematics, Sigma

References in corpus (5)

• Enumerative geometry of Calabi-Yau 4-folds
• Punctured logarithmic maps
• Stable maps to Looijenga pairs
• Curve counting in genus one: elliptic singularities & relative geometry
• The local-orbifold correspondence for simple normal crossings pairs

Cited by in corpus (8)

• Stable maps to Looijenga pairs
• On the log-local principle for the toric boundary
• Curve counting in genus one: elliptic singularities & relative geometry
• A case study of intersections on blowups of the moduli of curves
• The local-orbifold correspondence for simple normal crossings pairs
• Stable maps to Looijenga pairs: orbifold examples
• Sheaves of maximal intersection and multiplicities of stable log maps
• Enumerative geometry of surfaces and topological strings