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Regular version of the site

Working paper

On monodromy in families of elliptic curves over C

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the monodromy group has index at most 2m in SL(2,ℤ). By contrast, in any family of hyperelliptic curves of genus g≥3, the monodromy group is strictly less than Sp(2g,ℤ).  Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.