of Miranda moduli spaces of elliptic surfaces

We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $$ which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$ presented in terms of $$ , $$ on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other. Our approach exploits the natural $$ -action on Weierstrass curves and the identification of $$ -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $$ . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.