Divisors on the Moduli Space of Curves From Divisorial Conditions On Hypersurfaces

ABSTRACT

In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{{\mathcal{M}}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{{\mathcal{M}}_g}$, where the set of possible values of $g\in \{1,\dots ,N\}$ has density $\Omega (\hspace{0.17em}\text{log}\hspace{0.17em}{(N)}^{-0.087})$ for $N\gg 0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$.

KEYWORDS:

• Moduli space
• curves
• algebraic geometry
• equivariant intersection theory

AMS 2010 SUBJECT CLASSIFICATIONS::

• 14H10

Acknowledgments

The author would like to thank Joe Harris for introducing the author to moduli spaces of curves and for helpful discussions. The author would also like to thank Aleksei Kulikov for the reference [14] on the density of the candidate counterexamples to the Slope Conjecture, Anand Patel for the alternative proof of Theorem 1.3 presented, and Gavril Farkas for helpful comments.

Notes

1 There are some mild conditions required for $X{\times }^G\mathcal{P}$ to be a scheme, given in [6, Proposition 23]. Since products of general linear groups are special, they are always satisfied in our case.