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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Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method
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.