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On the monodromy group of the family of smooth plane curves

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arxiv 1610.04920 v1 pith:2FQFVG3D submitted 2016-10-16 math.GT math.AGmath.GR

classification math.GTmath.AGmath.GR
keywords groupclasscurvesmappingmonodromyplanedegreefamily
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We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an invariant known as a "n-spin structure" constrains the image in ways not predicted by previous work of Beauville. Second, we show that for degree d=5, our invariant is the only obstruction for a mapping class to be contained in the image. This requires combining the algebro-geometric work of L\"onne with Johnson's theory of the Torelli subgroup of the mapping class group.

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  1. Irrationality and monodromy for cubic threefolds

    math.AG 2019-08 accept novelty 6.0 of 10

    The monodromy homomorphism π1(M_{3,3}) → Sp(10;Z) for smooth cubic threefolds does not factor through the mapping class group of any genus five surface.

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