Non-smoothness of Moduli Spaces of Higher Genus Curves on Low Degree Hypersurfaces

Amal Mattoo; Matthew Hase-Liu

Non-smoothness of Moduli Spaces of Higher Genus Curves on Low Degree Hypersurfaces

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arxiv 2412.04618 v1 pith:GRYJUG2J submitted 2024-12-05 math.AG

Matthew Hase-Liu , Amal Mattoo This is my paper · ORCID

classification math.AG

keywords degreecurvesgenusmodulisingularsmootharbitrarybound

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abstract

We show that the moduli space of degree $e$ maps from smooth genus $g \ge 1$ curves to an arbitrary low degree smooth hypersurface is singular when $e$ is large compared to $g$. We also give a lower bound for the dimension of the singular locus.

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Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method
math.AG 2024-12 conditional novelty 8.0 of 10

For large ambient dimension n and curve degree e, the moduli space of genus g degree e maps into a smooth degree d hypersurface has at worst terminal singularities.

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