# Moscow Mathematical Journal

Volume 19, Issue 3, July–September 2019  pp. 597–613.

On Monodromy in Families of Elliptic Curves over ℂ

Authors:  Serge Lvovski (1)
Author institution:(1) National Research University Higher School of Economics, Russian Federation
Federal Scientific Centre Science Research Institute of System Analysis at Russian Academy of Science (FNP FSC SRISA RAS)

Summary:

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over $\mathbb{C}$ with a smooth base B for which the general fiber of the mapping J: B → 𝔸1 (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.

2010 Math. Subj. Class. 14D05, 14H52, 14J26.

Keywords: Monodromy, elliptic curve, hyperelliptic curve, jinvariant, braid, Del Pezzo surface.