On 3-folds having a holomorphic torus action with 6 fixed points

We study $3$-folds with an action of a algebraic torus $T$ and finite fixed point set. In particular, assuming the torus action has (exactly) $6$ fixed points we show that aside from Mori fibre spaces, the topology of such spaces is strongly restricted. For $T= \mathbb{C}^{*}$ we prove that there are two explicit infinite families plus a finite number of exceptional cases. For $T = \mathbb{C}^{*} \times \mathbb{C}^{*}$ there are $2$ exceptional cases which are described explicitly.