Here is a lemma regarding one-dimensional toric varieties.
Lemma
The normal one-dimensional toric varieties are \(\mathbb C^*\), \(\mathbb C\) and \(\mathbb P^1\). In particular, the only one dimensional complete toric variety is \(\mathbb P^1\).
Proof
If \(X\) is a normal one-dimensional toric variety, it corresponds to a fan in \(\mathbb R^1\). There are only three possibilities, the trivial fan, the fan with one ray and the fan with two rays.