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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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.