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

Abstract

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 $T={\mathbb{C}}^{\ast }$ we prove that there are two explicit infinite families plus a finite number of exceptional cases. For $T={\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$ there are $2$ exceptional cases which are described explicitly.

Exposition improved and minor corrections made. Main results unchanged. 16 pages, 0 figures