Schmitt, Johannes

Dimension Theory of the Moduli Space of Twisted $K$-Differentials

Doc. Math. 23, 871-894 (2018)
DOI: 10.25537/dm.2018v23.871-894


In this note we extend the dimension theory for the spaces $\widetilde{\mathcal{H}}_g^k(\mu)$ of twisted $k$-differentials defined by Farkas and Pandharipande in [G. Farkas and R. Pandharipande, J. Inst. Math. Jussieu 17, No. 3, 615--672 (2018; Zbl 06868654)] to the case $k>1$. In particular, we show that the intersection $\mathcal{H}_g^k(\mu)=\widetilde{\mathcal{H}}_g^k(\mu) \cap \mathcal{M}_{g,n}$ is a union of smooth components of the expected dimensions for all $k\geq 0$. We also extend a conjectural formula from [Zbl 06868654] for a weighted fundamental class of $\widetilde{\mathcal{H}}_g^k(\mu)$ and provide evidence in low genus. If true, this conjecture gives a recursive way to compute the cycle class $[\overline{\mathcal{H}}_g^k(\mu)]$ of the closure of $\mathcal{H}_g^k(\mu)$ for $k\geq 1,\mu$ arbitrary.

Mathematics Subject Classification

14H10, 30F30


strata of $k$-differentials, deformation theory, tautological classes, double ramification cycles


  • 1. M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Moeller. Compactification of strata of abelian differentials. Duke Math. J. , to appear. arxiv 1604.08834.
  • 2. M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Moeller. Strata of $k$-differentials. Algebr. Geom. , to appear. arxiv 1610.09238.
  • 3. Corentin Boissy. Connected components of the strata of the moduli space of meromorphic differentials. Comment. Math. Helv., 90(2):255--286, 2015. DOI 10.4171/CMH/353; zbl 1323.30060; MR3351745; arxiv 1211.4951.
  • 4. P. Belorousski and R. Pandharipande. A descendent relation in genus 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29(1):171--191, 2000. zbl 0981.81063; MR1765541; arxiv math/9803072.
  • 5. C. H. Clemens. A scrapbook of complex curve theory, volume 55 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2003. zbl 1030.14010; MR1946768.
  • 6. A. Dimca. Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004. zbl 1043.14003; MR2050072.
  • 7. D. Eisenbud and J. Harris. The Kodaira dimension of the moduli space of curves of genus $\geq 23$. Invent. Math., 90(2):359--387, 1987. DOI 10.1007/BF01388710; zbl 0631.14023; MR0910206.
  • 8. Gavril Farkas and Rahul Pandharipande. The moduli space of twisted canonical divisors. J. Inst. Math. Jussieu, 17(3):615--672, 2018. DOI 10.1017/S1474748016000128; zbl 06868654; MR3789183; arxiv 1508.07940.
  • 9. Q. Gendron. The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega\mathcal{M}_{g}$. Ann. Inst. Fourier. , to appear. DOI 10.5802/aif.3187; arxiv 1503.03338.
  • 10. J. Guéré. A generalization of the double ramification cycle via log-geometry. ArXiv e-prints, March 2016. arxiv 1603.09213.
  • 11. F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine. Double ramification cycles on the moduli spaces of curves. Publications mathématiques de l'IHÉS, 125(1):221--266, Jun 2017. DOI 10.1007/s10240-017-0088-x; zbl 1370.14029; MR3668650; arxiv 1602.04705.
  • 12. E. Lanneau. Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4), 41(1):1--56, 2008. DOI 10.24033/asens.2062; zbl 1161.30033; MR2423309; arxiv math/0506136.
  • 13. Martin Möller. Linear manifolds in the moduli space of one-forms. Duke Math. J., 144(3):447--487, 2008. DOI 10.1215/00127094-2008-041; zbl 1148.32007; MR2444303; arxiv math/0703145.
  • 14. Gabriele Mondello. On the cohomological dimension of the moduli space of riemann surfaces. Duke Math. J., 166(8):1463--1515, 06 2017. DOI 10.1215/00127094-0000004X; zbl 06754737; MR3659940; arxiv 1405.2608.
  • 15. H. Masur and J. Smillie. Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms. Comment. Math. Helv., 68(2):289--307, 1993. DOI 10.1007/BF02565820; zbl 0792.30030; MR1214233.
  • 16. A. Pixton. Double ramification cycles and tautological relations on $\overline{\mathcal{M}}_{g,n}$. preprint, 2014.
  • 17. A. Polishchuk. Moduli spaces of curves with effective $r$-spin structures. In Gromov-Witten theory of spin curves and orbifolds, volume 403 of Contemp. Math., pages 1--20. Amer. Math. Soc., Providence, RI, 2006. zbl 1112.14028; MR2234882; arxiv math/0309217.
  • 18. R. Pandharipande, A. Pixton, and D. Zvonkine. Relations on $\overline{\mathcal M}_{g,n}$ via $3$-spin structures. J. Amer. Math. Soc., 28(1):279--309, 2015. zbl 1315.14037; MR3264769; arxiv 1303.1043.
  • 19. Kurt Strebel. Quadratic differentials, volume 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984. zbl 0547.30001; MR0743423.
  • 20. N. Tarasca. Double total ramifications for curves of genus 2. Int. Math. Res. Not. IMRN, (19):9569--9593, 2015. DOI 10.1093/imrn/rnu228; zbl 1348.14077; MR3431602; arxiv 1401.3057.
  • 21. William A. Veech. The teichmuller geodesic flow. Annals of Mathematics, 124(3):441--530, 1986. DOI 10.2307/2007091; zbl 0658.32016; MR0866707.
  • 22. William A. Veech. Flat surfaces. American Journal of Mathematics, 115(3):589--689, 1993. DOI 10.2307/2375075; zbl 0803.30037; MR1221838.
  • 23. Scott A. Wolpert. Infinitesimal deformations of nodal stable curves. Advances in Mathematics, 244(Supplement C):413 -- 440, 2013. DOI 10.1016/j.aim.2013.05.008; zbl 1290.14019; MR3077878; arxiv 1204.3680.


Schmitt, Johannes
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland