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Статья

On monodromy in families of elliptic curves over C

Moscow Mathematical Journal. 2019. Vol. 19. No. 3. P. 597-613.

We show that if we are given a smooth non-isotrivial family of curves of genus 1 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.