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Tevelev degrees and Hurwitz moduli spaces

Published online by Cambridge University Press:  03 December 2021

A. CELA
Affiliation:
Department of Mathematics, ETH Zurich Office: HG J 14.4 Rämistrasse 101 8092 ZurichSwitzerland e-mails: alessio.cela@math.ethz.ch
R. PANDHARIPANDE
Affiliation:
Department of Mathematics, ETH Zurich Office: HG G 55 Rämistrasse 101 8092 ZurichSwitzerland e-mails: rahul@math.ethz.ch
J. SCHMITT
Affiliation:
Institute for Mathematics, University of Zurich Office: Y27K42 Winterthurerstrasse 190 8057 ZürichSwitzerland e-mail: johannes.schmitt@math.uzh.ch

Abstract

We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Arkani-Hamed, N., Bourjaily, J., Cachazo, F., Postnikov, A., and Trnka, J.. On-shell structures of MHV amplitudes beyond the planar limit. J. High Energy Physics 6 (2015), 179.CrossRefGoogle Scholar
Bae, Y. and Schmitt, J.. Chow rings of stacks of prestable curves I. arXiv:2012.09887.Google Scholar
Belorousski, P. and Pandharipande, R.. A descendent relation in genus 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000), 171191.Google Scholar
Bouchard, V. and MariÑo, M.. Hurwitz numbers, matrix models and enumerative geometry in Hodge theory to integrability and TQFT tt*-geometry. Proc. Sympos. Pure Math. 78 (2008), 263–283.Google Scholar
Buch, A. and Pandharipande, R.. Tevelev degrees in Gromov–Witten theory. In preparation.Google Scholar
Castelnuovo, G.. Numero delle involuzioni razionali giacenti sopra una curva di dato genere. Rendiconti R. Accad. Lincei 5 (1889), 130–133.Google Scholar
Cavalieri, R.. Markwig, H. and Ranganathan, D.. In preparation.Google Scholar
Comtet, L.. Advanced combinatorics, the art of finite and infinite expansions. (D. Reidel, Dordrecht and Boston, 1974).Google Scholar
Dijkgraaf, R.. Mirror symmetry and elliptic curves. The Moduli Space of Curves, Dijkgraaf, R., C. Faber, G. van der Geer (editors). Progress in Mathematics, 129 (Birkhäuser, 1995).CrossRefGoogle Scholar
Deutsch, E.. Dyck path enumeration. Discrete Math. 204 (1999), 167202.CrossRefGoogle Scholar
Ekedahl, T.. Lando, S., Shapiro, M. and Vainshtein, A.. Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146 (2001), 297327.Google Scholar
Faber, C. and Pandharipande, R.. Relative maps and tautological classes. JEMS 7 (2005), 1349.Google Scholar
Farkas, G. and Lian, C.. Linear series on general curves with prescribed incidence conditions. arXiv:2105.09340.Google Scholar
Graber, T. and Pandharipande, R.. Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J. 51 (2003), 93109.CrossRefGoogle Scholar
Harris, J. and Mumford, D.. On the Kodaira dimension of the moduli space of curves With an appendix by William Fulton. Invent. Math. 67 (1982), 23–88.Google Scholar
Hurwitz, A.. Ueber die Anzahl der Riemann’schen Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 55 (1901), 53–66.Google Scholar
Janda, F.. Pandharipande, R., Pixton, A. and Zvonkine, D.. Double ramification cycles on the moduli spaces of curves. Publ. Math. Inst. Hautes Études Sci. 125 (2017), 221–266.CrossRefGoogle Scholar
Lian, C.. Non-tautological Hurwitz cycles. arXiv:2101.11050.Google Scholar
Lian, C.. appeared in Selecta Math. (N.S.) 27, Article Nr. 96 (2021). arXiv:2011.11565.Google Scholar
Okounkov, A. and Pandharipande, R.. Gromov–Witten theory, Hurwitz numbers, and completed cycles. Ann. of Math. 163 (2006), 517–560.Google Scholar
Okounkov, A. and Pandharipande, R.. Gromov–Witten theory, Hurwitz numbers, and matrix models. Proceedings of Algebraic Geometry (Seattle 2005) Proc. Sympos. Pure Math. 80, Part 1, 324–414.Google Scholar
Pandharipande, R.. A geometric construction of Getzler’s elliptic relation. Math. Ann. 313 (1999), 715–729.Google Scholar
Pandharipande, R. and Tseng, H.-H.. Higher genus Gromov–Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ and CohFTs associated to local curves. Forum Math. Pi 7 (2019).CrossRefGoogle Scholar
SageMath, S.. The Sage Mathematics Software System (Version 9.1), The Sage Developers (2021) https://www.sagemath.org.Google Scholar
Schmitt, J. and van Zelm, J.. Intersections of loci of admissible covers with tautological classes. Selecta Math. 26 (2020).CrossRefGoogle Scholar
Scott, B.. Number of Dyck paths that touch the diagonal exactly k times, Mathematics Stack Exchange. https://math.stackexchange.com/questions/1900928/number-of-dyck-paths-that-touch-the-diagonal-exactly-k-times.Google Scholar
Stanley, R.. Enumerative Combinatorics, vol 2 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
Tevelev, J.. Scattering amplitudes of stable curves, arXiv:2007.03831.Google Scholar
van Zelm, J.. Nontautological bielliptic cycles. Pacific J. Math. 294 (2018), 495–504.CrossRefGoogle Scholar

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