Divisors on the moduli space of curves from divisorial conditions on\n hypersurfaces

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

paper · pdf · doi:10.48550/arxiv.1901.11154

openalex publication_date 2019/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we extend work of Farkas and Rim 'anyi on applying quadric rank\nloci to finding divisors of small slope on the moduli space of curves by\ninstead considering all divisorial conditions on the hypersurfaces of a fixed\ndegree containing a projective curve. This gives rise to a large family of\nvirtual divisors on \\M_g. We determine explicitly which\nof these divisors are candidate counterexamples to the Slope Conjecture. The\npotential counterexamples exist on \\M_g, where the set of\npossible values of g\∈ 1,\…,N has density \Ω(\log(N)^-0.087)\nfor N>>0. Furthermore, no divisorial condition defined using hypersurfaces of\ndegree greater than 2 give counterexamples to the Slope Conjecture, and every\ndivisor in our family has slope at least 6+\(8)/(g+1).\n

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