Variations on twists of tuples of hyperelliptic curves and related results

Let f∈Q[x] be a square-free polynomial of degree ⩾3 and m⩾3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D1∈Q(u,v,w) such that the Jacobians of the curvesC1:D1y2=f(x),C2:y2=D1xm+b,C3:y2=D1xm+c, have all positive ranks over Q(u,v,w). Similarly, we prove that there exists a function D2∈Q(u,v,w) such that the Jacobians of the curvesC1:D2y2=f(x),C2:y2=D2xm+b,C3:y2=xm+cD2, have all positive ranks over Q(u,v,w). Moreover, if f(x)=xm+a for some a∈Z∖{0}, we prove the existence of a function D3∈Q(u,v,w) such that the Jacobians of the curvesC1:y2=D3xm+a,C2:y2=D3xm+b,C3:y2=xm+cD3, 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 yp=xm(x+a) and yp=xm(a−x)k for a prime p and 0