Two-dimensional families of hyperelliptic Jacobians with big monodromy

Let K be a global field of characteristic different from $2$ and u(x)∈ K[x] be an irreducible polynomial of even degree 2g≥ 6 whose Galois group over K is either the full symmetric group S2g or the alternating group A2g. We describe explicitly how to choose (infinitely many) pairs of distinct t₁, t₂ ∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y²= (x-t₁)(x-t₂))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy.