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Tangent curves to degenerating hypersurfaces

  • Lawrence Jack Barrott and Navid Nabijou EMAIL logo

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.

Award Identifier / Grant number: EP/R009325/1

Funding statement: Lawrence Jack Barrott was supported by the National Centre for Theoretical Sciences Taipei and Boston College. Navid Nabijou was supported by EPSRC grant EP/R009325/1 and the Herchel Smith Fund.

Acknowledgements

It is a pleasure to thank Pierrick Bousseau and Tim Gräfnitz for many inspiring discussions. We also thank Dan Abramovich, Michel van Garrel, Tom Graber, Mark Gross, Sanghyeon Lee, Dhruv Ranganathan and Helge Ruddat for helpful conversations. We are grateful to the anonymous referee for numerous helpful suggestions and expositional advice, as well as the observation that our logarithmically singular degeneration formula might have applications beyond hypersurface degenerations. Parts of this work were carried out during research visits at the National Centre for Theoretical Sciences Taipei, the University of Glasgow, the Mathematisches Forschungsinstitut Oberwolfach and Boston College, and it is a pleasure to thank these institutions for hospitality and financial support.

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Received: 2021-03-22
Revised: 2022-06-20
Published Online: 2022-10-27
Published in Print: 2022-12-01

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