Abstract
We show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on {overline{{mathcal {M}}}}_{g,N}, assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.
Highlights
1.1 Tautological classes on moduli spaces of curvesThe Chow A∗(Mg,n) and cohomology H ∗(Mg.n) rings of moduli spaces of stable pointed curves are central objects of enumerative geometry
Many cohomology classes on moduli spaces of curves arising in geometry turn out to be tautological
Step 1 (h = 1, g + m2 = 12) : We pull back the cycle Hg/1,d,(m2)2 ∈ H ∗(Mg,2m2 ) to the boundary stratum M1,11 × M1,11 obtained by gluing g − 1 pairs of nodes on the two elliptic components together
Summary
The Chow A∗(Mg,n) and cohomology H ∗(Mg.n) rings of moduli spaces of stable pointed curves are central objects of enumerative geometry. While both objects are extremely complicated and likely impossible to understand completely, Mumford [12] initiated a study of certain tautological classes on Mg,n that appear in many natural geometric situations and are largely computable in practice. Additive generators for the tautological ring and formulas for their intersections may be given combinatorially, see [6, Appendix A]. Many cohomology classes on moduli spaces of curves arising in geometry turn out to be tautological. Using techniques of Gromov-Witten theory, Faber-Pandharipande [3] show that loci of curves admitting maps to P1 with prescribed ramification profiles are tautological.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
