On monodromy in families of elliptic curves over \mathbb C

2017/05/05 by Lvovski, Serge
#14D05 #14H52 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.02129

Abstract

We show that if we are given a smooth non-isotrivial family of elliptic curves over~\mathbb C with a smooth base~B for which the general fiber of the mapping J\colon B→\mathbb A¹ (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H¹(⋅,\mathbb Z) of the fibers) coincides with SL(2,\mathbb Z); if the general fiber has m≥2 connected components, then the monodromy group has index at most~2m in SL(2,\mathbb Z). By contrast, in any family of hyperelliptic curves of genus g≥3, the monodromy group is strictly less than Sp(2g,\mathbb Z). Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

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