Variations on twists of tuples of hyperelliptic curves and related results

Let $f\in\Q[x]$ be a square-free polynomial of degree $\geq 3$ and $m\geq 3$ be an odd positive integer. Based on our earlier investigations we prove that there exists a function $D_{1}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{1}y^2=f(x),\quad C_{2}:\;y^2=D_{1}x^m+b,\quad C_{3}:\;y^2=D_{1}x^m+c, \end{equation*} have all positive ranks over $\Q(u,v,w)$. Similarly, we prove that there exists a function $D_{2}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{2}y^2=h(x),\quad C_{2}:\;y^2=D_{2}x^m+b,\quad C_{3}:\;y^2=x^m+cD_{2}, \end{equation*} have all positive ranks over $\Q(u,v,w)$. Moreover, if $f(x)=x^m+a$ for some $a\in\Z\setminus\{0\}$, we prove the existence of a function $D_{3}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;y^2=D_{3}x^{m}+a,\quad C_{2}:\;y^2=D_{3}x^m+b,\quad C_{3}:\;y^2=x^m+cD_{3}, \end{equation*} have all positive ranks over $\Q(u,v,w)$. We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves $y^p=x^{m}(x+a)$ and $y^p=x^{m}(a-x)^{k}$ for a prime $p$ and $0<m<p-2$ and $k<p$ and apply our result in order to prove the existence of a function $D\in\Q(u,v,w,t)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;Dy^p=x^m(x+a),\quad Dy^p=x^m(x+b) \end{equation*} have both positive rank over $\Q(u,v,w,t)$.