On monodromy in families of elliptic curves over $\mathbb{C}$

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 $J\colon B\to\mathbb{A}^1$ (assigning $j$-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on $H^1(\cdot,\mathbb{Z})$ of the fibers) coincides with $\mathrm{SL}(2,\mathbb{Z})$; if the general fiber has $m\ge2$ connected components, then the monodromy group has index at most $2m$ in $\mathrm{SL}(2,\mathbb{Z})$. By contrast, in any family of hyperelliptic curves of genus $g\ge3$, the monodromy group is strictly less than $\mathrm{Sp}(2g,\mathbb{Z})$. Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.