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 $J\text{\hspace{0.17em}⁣}:B\to {\mathbb{A}}^1$ (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{S}\mathrm{L}$(2,\mathbb Z)$;ifthegeneralfiberhas$m\ge2$ connected components, then the monodromy group has index at most~$2m$ in $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. By contrast, in \emph{any} family of hyperelliptic curves of genus $g\ge 3$, the monodromy group is strictly less than $\mathrm{S}\mathrm{p}(2g,\mathbb{Z})$. Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.