# 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* → 𝔸^{1} (assigning *j*-invariant of
the fiber to a point) is connected, then the monodromy group of the
family (acting on *H*^{1}(·,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 2*m* in SL(2,ℤ). By
contrast, in *any* family of hyperelliptic curves of genus
*g*≥3, the monodromy group is strictly less than Sp(2*g*,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.

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