2019/01/30 by Dennis Tseng, Tseng, Dennis
Mathematics · #14H15 #55N91 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #advanced mathematical theories #math.AG #msc:14H15 #msc:55N91
paper · pdf · doi:10.48550/arxiv.1901.11154
Revised according to referee comments, to appear in Experimental Mathematics
openalex publication_date 2019/01/30 · arxiv created 2021/10/05 · arxiv updated 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Mg. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on Mg, where the set of possible values of g∈ \1,…,N\ has density Ω(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+(8)/(g+1).