Double ramification cycles on the moduli spaces of curves

Janda, F.; Pandharipande, R.; Pixton, A.; Zvonkine, D.

Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 221-266.

Curves of genus $g$ which admit a map to ${\mathbf{P}}^1$ with specified ramification profile $\mu$ over $0\in {\mathbf{P}}^1$ and $\nu$ over $\mathrm{\infty }\in {\mathbf{P}}^1$ define a double ramification cycle ${\mathsf{DR}}_g(\mu ,\nu )$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.

The cycle ${\mathsf{DR}}_g(\mu ,\nu )$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for ${\mathsf{DR}}_g(\mu ,\nu )$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.

When $\mu =\nu =\mathrm{\varnothing }$, the formula for double ramification cycles expresses the top Chern class ${\lambda }_g$ of the Hodge bundle of ${\bar{\mathcal{M}}}_g$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

[1.] Abramovich, D.; Graber, T.; Vistoli, A. Gromov-Witten theory of Deligne-Mumford stacks, Am. J. Math., Volume 130 (2008), pp. 1337-1398 | DOI | MR | Zbl

[2.] Bryan, J.; Pandharipande, R. The local Gromov-Witten theory of curves, J. Am. Math. Soc., Volume 21 (2008), pp. 101-136 | DOI | MR | Zbl

[3.] Buryak, A.; Shadrin, S.; Spitz, L.; Zvonkine, D. Integrals of $\psi$-classes over double ramification cycles, Am. J. Math., Volume 137 (2015), pp. 699-737 | DOI | MR | Zbl

[4.] Cavalieri, R.; Marcus, S.; Wise, J. Polynomial families of tautological classes on ${\mathcal{M}}_{g,n}^{\mathsf{rt}}$, J. Pure Appl. Algebra, Volume 216 (2012), pp. 950-981 | DOI | MR | Zbl

[5.] Chiodo, A. Stable twisted curves and their $r$-spin structures, Ann. Inst. Fourier, Volume 58 (2008), pp. 1635-1689 | DOI | Numdam | MR | Zbl

[6.] Chiodo, A. Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math., Volume 144 (2008), pp. 1461-1496 | DOI | MR | Zbl

[7.] Chiodo, A.; Zvonkine, D. Twisted Gromov-Witten $r$-spin potential and Givental’s quantization, Adv. Theor. Math. Phys., Volume 13 (2009), pp. 1335-1369 | DOI | MR | Zbl

[8.] E. Clader and F. Janda, Pixton’s double ramification cycle relations, | arXiv

[9.] Faber, C. A conjectural description of the tautological ring of the moduli space of curves, Moduli of Curves and Abelian Varieties (1999), pp. 109-129 | DOI

[10.] Faber, C.; Pandharipande, R. Hodge integrals and Gromov-Witten theory, Invent. Math., Volume 139 (2000), pp. 173-199 | DOI | MR | Zbl

[11.] Faber, C.; Pandharipande, R. Relative maps and tautological classes, J. Eur. Math. Soc., Volume 7 (2005), pp. 13-49 | DOI | MR | Zbl

[12.] Faber, C.; Pandharipande, R. Tautological and non-tautological cohomology of the moduli space of curves, Handbook of Moduli, vol. I (2013), pp. 293-330

[13.] G. Farkas and R. Pandharipande, The moduli space of twisted canonical divisors, with Appendix by F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, J. Inst. Math. Jussieu (2017, to appear), doi:, | arXiv | DOI

[14.] Getzler, E.; Pandharipande, R. Virasoro constraints and the Chern classes of the Hodge bundle, Nucl. Phys. B, Volume 530 (1998), pp. 701-714 | DOI | MR | Zbl

[15.] Graber, T.; Pandharipande, R. Localization of virtual classes, Invent. Math., Volume 135 (1999), pp. 487-518 | DOI | MR | Zbl

[16.] Graber, T.; Pandharipande, R. Constructions of nontautological classes on moduli spaces of curves, Mich. Math. J., Volume 51 (2003), pp. 93-109 | DOI | MR | Zbl

[17.] Graber, T.; Vakil, R. Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J., Volume 130 (2005), pp. 1-37 | DOI | MR | Zbl

[18.] Grushevsky, S.; Zakharov, D. The double ramification cycle and the theta divisor, Proc. Am. Math. Soc., Volume 142 (2014), pp. 4053-4064 | DOI | MR | Zbl

[19.] Hain, R. Normal functions and the geometry of moduli spaces of curves, Handbook of Moduli, vol. I (2013), pp. 527-578

[20.] Heller, I.; Tompkins, C. An extension of a theorem of Dantzig’s, Linear Inequalities and Related Systems (1956), pp. 247-254

[21.] F. Janda, Relations on ${\bar{\mathcal{M}}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbf{P}}^1$, Alg. Geom., (2017, to appear), | arXiv

[22.] F. Janda, R. Pandharipande, A. Pixton and D. Zvonkine, in preparation.

[23.] Jarvis, T. J. Geometry of the moduli of higher spin curves, Int. J. Math., Volume 11 (2000), pp. 637-663 | MR | Zbl

[24.] Johnson, P.; Pandharipande, R.; Tseng, H.-H. Abelian Hurwitz-Hodge integrals, Mich. Math. J., Volume 60 (2011), pp. 171-198 (arXiv:0803.0499) | DOI | MR | Zbl

[25.] Li, J. A degeneration formula of GW invariants, J. Differ. Geom., Volume 60 (2002), pp. 199-293 | DOI | MR | Zbl

[26.] J. Li, Lecture notes on relative GW-invariants, http://users.ictp.it/~pub_off/lectures/lns019/Jun_Li/Jun_Li.pdf.

[27.] S. Marcus and J. Wise, Stable maps to rational curves and the relative Jacobian, | arXiv

[28.] Mumford, D. Towards an enumerative geometry of the moduli space of curves, Arithmetic and Geometry (1983), pp. 271-328 | DOI

[29.] Okounkov, A.; Pandharipande, R. Gromov-Witten theory, Hurwitz numbers, and completed cycles, Ann. Math., Volume 163 (2006), pp. 517-560 | DOI | MR | Zbl

[30.] Okounkov, A.; Pandharipande, R. Quantum cohomology of the Hilbert scheme of points of the plane, Invent. Math., Volume 179 (2010), pp. 523-557 | DOI | MR | Zbl

[31.] R. Pandharipande and A. Pixton, Relations in the tautological ring of the moduli space of curves, | arXiv

[32.] Pandharipande, R.; Pixton, A.; Zvonkine, D. Relations on ${\bar{\mathcal{M}}}_{g,n}$ via 3-spin structures, J. Am. Math. Soc., Volume 28 (2015), pp. 279-309 | DOI | MR | Zbl

[33.] A. Pixton, Conjectural relations in the tautological ring of ${\bar{\mathcal{M}}}_{g,n}$, | arXiv

[34.] A. Pixton, Double ramification cycles and tautological relations on ${\bar{\mathcal{M}}}_{g,n}$, available from the author.

[35.] A. Pixton, On combinatorial properties of the explicit expression for double ramification cycles, in preparation.

[36.] Riemann, B. Theorie der Abel’schen Functionen, J. Reine Angew. Math., Volume 54 (1857), pp. 115-155 | DOI | MR

[37.] Stanley, R. Enumerative Combinatorics: Vol I (1999) | DOI

[38.] van der Geer, G. Cycles on the moduli space of abelian varieties, Moduli of Curves and Abelian Varieties (1999), pp. 65-89 | DOI