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Tangent curves to degenerating hypersurfaces

  • Lawrence Jack Barrott and Navid Nabijou EMAIL logo


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.


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.


[1] D. Abramovich and Q. Chen, Stable logarithmic maps to Deligne–Faltings pairs II, Asian J. Math. 18 (2014), no. 3, 465–488. 10.4310/AJM.2014.v18.n3.a5Search in Google Scholar

[2] D. Abramovich, Q. Chen, D. Gillam, Y. Huang, M. Olsson, M. Satriano and S. Sun, Logarithmic geometry and moduli, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM) 24, International Press, Somerville (2013), 1–61. Search in Google Scholar

[3] D. Abramovich, Q. Chen, M. Gross and B. Siebert, Decomposition of degenerate Gromov–Witten invariants, Compos. Math. 156 (2020), no. 10, 2020–2075. 10.1112/S0010437X20007393Search in Google Scholar

[4] D. Abramovich, Q. Chen, M. Gross and B. Siebert, Punctured logarithmic maps, preprint (2020), Search in Google Scholar

[5] D. Abramovich, Q. Chen, S. Marcus, M. Ulirsch and J. Wise, Skeletons and fans of logarithmic structures, Nonarchimedean and tropical geometry, Simons Symp., Springer, Cham (2016), 287–336. 10.1007/978-3-319-30945-3_9Search in Google Scholar

[6] D. Abramovich and J. Wise, Birational invariance in logarithmic Gromov–Witten theory, Compos. Math. 154 (2018), no. 3, 595–620. 10.1112/S0010437X17007667Search in Google Scholar

[7] L. Battistella, N. Nabijou and D. Ranganathan, Curve counting in genus one: Elliptic singularities and relative geometry, Algebr. Geom. 8 (2021), no. 6, 637–679. 10.14231/AG-2021-020Search in Google Scholar

[8] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. 10.1007/s002220050136Search in Google Scholar

[9] K. A. Behrend, On the de Rham cohomology of differential and algebraic stacks, Adv. Math. 198 (2005), no. 2, 583–622. 10.1016/j.aim.2005.05.025Search in Google Scholar

[10] N. Borne and A. Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), no. 3–4, 1327–1363. 10.1016/j.aim.2012.06.015Search in Google Scholar

[11] P. Bousseau, A proof of N.Takahashi’s conjecture for ( 2 , E ) and a refined sheaves/Gromov–Witten correspondence, preprint (2019), Search in Google Scholar

[12] R. Cavalieri, M. Chan, M. Ulirsch and J. Wise, A moduli stack of tropical curves, Forum Math. Sigma 8 (2020), Paper No. e23. 10.1017/fms.2020.16Search in Google Scholar

[13] Q. Chen, Stable logarithmic maps to Deligne–Faltings pairs I, Ann. of Math. (2) 180 (2014), no. 2, 455–521. 10.4007/annals.2014.180.2.2Search in Google Scholar

[14] J. Choi, M. van Garrel, S. Katz and N. Takahashi, Sheaves of maximal intersection and multiplicities of stable log maps, Selecta Math. (N. S.) 27 (2021), no. 4, Paper No. 61. 10.1007/s00029-021-00671-0Search in Google Scholar

[15] D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry, Math. Surveys Monogr. 68, American Mathematical Society, Providence 1999. 10.1090/surv/068Search in Google Scholar

[16] S. Felten, M. Filip and H. Ruddat, Smoothing toroidal crossing spaces, Forum Math. Pi 9 (2021), Paper No. e7. 10.1017/fmp.2021.8Search in Google Scholar

[17] A. Gathmann, Relative Gromov–Witten invariants and the mirror formula, Math. Ann. 325 (2003), no. 2, 393–412. 10.1007/s00208-002-0345-1Search in Google Scholar

[18] W. D. Gillam, Logarithmic flatness, preprint (2016), Search in Google Scholar

[19] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. 10.1007/s002220050293Search in Google Scholar

[20] T. Gräfnitz, Tropical correspondence for smooth del Pezzo log Calabi–Yau pairs, preprint (2020),; to appear in J. Algebraic Geom. 10.1090/jag/794Search in Google Scholar

[21] M. Gross, R. Pandharipande and B. Siebert, The tropical vertex, Duke Math. J. 153 (2010), no. 2, 297–362. 10.1215/00127094-2010-025Search in Google Scholar

[22] M. Gross and B. Siebert, Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. 10.1090/S0894-0347-2012-00757-7Search in Google Scholar

[23] M. Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island 1994), Progr. Math. 129, Birkhäuser, Boston (1995), 335–368. 10.1007/978-1-4612-4264-2_12Search in Google Scholar

[24] Y.-P. Lee and R. Pandharipande, A reconstruction theorem in quantum cohomology and quantum K-theory, Amer. J. Math. 126 (2004), no. 6, 1367–1379. 10.1353/ajm.2004.0049Search in Google Scholar

[25] J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. 10.4310/jdg/1090351102Search in Google Scholar

[26] B. H. Lian, K. Liu and S.-T. Yau, Mirror principle. III, Asian J. Math. 3 (1999), no. 4, 771–800. 10.4310/AJM.1999.v3.n4.a4Search in Google Scholar

[27] C. Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), no. 2, 201–245. 10.1090/S1056-3911-2011-00606-1Search in Google Scholar

[28] C. Manolache, Virtual push-forwards, Geom. Topol. 16 (2012), no. 4, 2003–2036. 10.2140/gt.2012.16.2003Search in Google Scholar

[29] D. Maulik and R. Pandharipande, A topological view of Gromov-Witten theory, Topology 45 (2006), no. 5, 887–918. 10.1016/ in Google Scholar

[30] W. A. Nizioł, Toric singularities: Log-blow-ups and global resolutions, J. Algebraic Geom. 15 (2006), no. 1, 1–29. 10.1090/S1056-3911-05-00409-1Search in Google Scholar

[31] A. Ogus, Lectures on logarithmic algebraic geometry, Cambridge Stud. Adv. Math. 178, Cambridge University, Cambridge 2018. 10.1017/9781316941614Search in Google Scholar

[32] A. Okounkov and R. Pandharipande, Gromov–Witten theory, Hurwitz numbers, and matrix models, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 325–414. 10.1090/pspum/080.1/2483941Search in Google Scholar

[33] M. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), no. 5, 747–791. 10.1016/j.ansens.2002.11.001Search in Google Scholar

[34] M. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. 10.1007/s00208-005-0707-6Search in Google Scholar

[35] R. Pandharipande and A. Pixton, Gromov–Witten/Pairs correspondence for the quintic 3-fold, J. Amer. Math. Soc. 30 (2017), no. 2, 389–449. 10.1090/jams/858Search in Google Scholar

[36] Z. Ran, The number of unisecant rational cubics to a plane cubic, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 196, 487–489. 10.1093/qmathj/49.4.487Search in Google Scholar

[37] D. Ranganathan, Logarithmic Gromov–Witten theory with expansions, preprint (2019),; to appear in Algebr. Geom. 10.14231/AG-2022-022Search in Google Scholar

[38] D. Ranganathan, K. Santos-Parker and J. Wise, Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol. 23 (2019), no. 7, 3315–3366. 10.2140/gt.2019.23.3315Search in Google Scholar

[39] T. Stacks Project Authors, Stacks project,, 2018. Search in Google Scholar

[40] N. Takahashi, Curves in the complement of a smooth plane cubic whose normalizations are 𝔸 1 , preprint (1996), Search in Google Scholar

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

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