# Divisors on the moduli space of curves from divisorial conditions on hypersurfaces

## TL;DR

This paper explores divisors on the moduli space of curves arising from hypersurfaces, identifying potential counterexamples to the Slope Conjecture and establishing bounds on divisor slopes.

## Contribution

It extends previous work by considering all divisorial conditions on hypersurfaces, providing explicit candidate counterexamples and slope bounds.

## Key findings

- Potential counterexamples to the Slope Conjecture exist with density related to logarithmic growth.
- Divisorial conditions using hypersurfaces of degree greater than 2 do not produce counterexamples.
- All divisors in the family have slopes at least 6 + 8/(g+1).

## Abstract

In this note, we extend work of Farkas and Rim\'anyi 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 $g\in \{1,\ldots,N\}$ has density $\Omega(\log(N)^{-0.087})$ for $N>>0$. 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 $6+\frac{8}{g+1}$.