# Fibred algebraic surfaces and commutators in the Symplectic group

## TL;DR

This paper investigates minimal critical points and singular fibers in non-isotrivial surface fibrations over genus 1 curves, and characterizes certain products of transvections as commutators in the symplectic group.

## Contribution

It constructs explicit fibrations with minimal singular fibers and describes which transvection products are commutators in the symplectic group.

## Key findings

- Constructed a fibration with one singular fiber with four nodes.
- Explicitly characterized products of transvections that are commutators.
- Determined minimal critical points for specific surface fibrations.

## Abstract

We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.