Tangent curves to degenerating hypersurfaces

TL;DR

This paper investigates how rational tangent curves to hypersurfaces behave during degenerations, extending logarithmic Gromov-Witten theory to produce refined invariants and applying tropical and localization techniques for explicit computations.

Contribution

It extends the degeneration formula in logarithmic Gromov-Witten theory to singular settings and refines invariants to capture tangent curve behavior during hypersurface degenerations.

Findings

Refined logarithmic Gromov-Witten invariants for degenerating hypersurfaces.

Explicit computations for a degenerated smooth plane cubic using tropical methods.

Classical descriptions of tangent curve degenerations derived from logarithmic Gromov-Witten theory.

Abstract

We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov-Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov-Witten theory.