Fundamental groups of moduli stacks of smooth Weierstrass fibrations

TL;DR

This paper extends classical results on the fundamental group of elliptic curves to moduli stacks of smooth Weierstrass fibrations over complex projective spaces, providing finite presentations and generalizing SL_2(Z).

Contribution

It offers new finite presentations for the fundamental groups of moduli stacks of Weierstrass curves, extending classical elliptic curve results to higher-dimensional bases.

Findings

Finite presentations for fundamental groups of moduli stacks of Weierstrass curves.

Generalization of SL_2(Z) to higher-dimensional base spaces.

Methodology based on involution and fixed loci analysis.

Abstract

We give finite presentations for the fundamental group of moduli stacks of smooth Weierstrass curves over complex projective space P^n which extend the classical result for elliptic curves to positive dimensional base. We thus get natural generalisations of SL_2(Z) and pave the way to understanding the fundamental group of moduli stacks of elliptic surfaces in general. Our approach exploits the natural involution on Weierstrass curves and the identification of its fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over P^n. 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, which were sucessfully elaborated in mathAG/0602371.