About the equivalence of divisor classes on hyperelliptic curves and a   quotient of linear forms by an algebraic group action
  For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor classes. This structure reduces the question of rationality of divisor classes to rationality of polynomials which is more easy to control. 
Victor Gonzalo Lopez Neumann