On monodromy in families of elliptic curves over mathbb C

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 $J\colon B\to\mathbb A^1$ (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 \emph{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.