Level Structures on the Weierstrass Family of Cubics

Let W →  2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 2 . Since W →  2 is the versal deformation space o...

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Bibliographic Details
Published in:Communications in algebra 2007-04, Vol.35 (4), p.1249-1261
Main Authors: Bernstein, Mira, Tuffley, Christopher
Format: Article
Language:English
Subjects:
Branched covers of 3-manifolds
Level structure
Primary 14D05
Secondary 14H20, 57M12
Versal deformation space of a cusp
Weierstrass curves
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Summary:Let W →  2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 2 . Since W →  2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S 3 with monodromy in SL 2 (ℤ/N).
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870601142256