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.
On the monodromy group of the family of smooth plane curves
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abstract
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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Irrationality and monodromy for cubic threefolds
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.