Abstract
We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over {mathbf {P}}^1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of SL_2{{mathbb {Z}}} presented in terms of Bigg (begin{array}{ll} 1&{}1 0&{}1end{array} Bigg ), Bigg (begin{array}{ll} 1&{}0 {-1}&{}1end{array} Bigg ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural {mathbb {Z}}_2-action on Weierstrass curves and the identification of {mathbb {Z}}_2-fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over {{mathbf {P}}}^1. 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.
Highlights
Our primary objects are curves on the ruled surface Xd = P OP1 (d) ⊕ OP1 in the linear system |3σ0|, where σ0 denotes the divisor on Xd defined by the zero section of OP1 (d)
On Vd we introduce coordinates u◦, uν, uν, uν with respect to the monomial basis such that the tautological hypersurface Hd is given by the vanishing of u◦y3 +
Dd = u ∈ PVd the fibre H|u is singular which is the union of the hyperplane {u◦ = 0} and the projective dual of the weighted projective plane Pd2,1,1 given as the image of Xd under the projective morphism defined by the base point free linear system |3σ0|
Summary
Which is given as the direct product of the group of linear base and a torus C∗ acting on the coordinates uν, uν see π1(Ud ) and derive the orbifold fundamental group of Md from a homotopy exact sequence In this way we are able to generalise to base dimension one the old result giving the orbifold fundamental group of the moduli space of elliptic curves, which is naturally the moduli space of smooth Weierstrass fibrations over the point. The complement of the discriminant in the unfolding base was shown to have fundamental group generated as in the theorem but with relations (i)–(iii) only, [7], in terms of the graph n, which incidentally is a distinguished Dynkin graph associated to the singularity of f. (iii) A suitable relation of smooth elliptic fibrations with section to Weierstrass fibrations with mild singularities is needed for any progress on geometric monodromy
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