Complete fan

A fan is complete when the union of all its cones is the whole ambient real vector space. A toric variety is proper exactly when its fan is complete.

For a cone $\tau \in \Sigma$, its star is the fan in the quotient by $\mathrm{span}(\tau )$ formed from the images of all cones $\sigma$ containing $\tau$ as a face.

In dimension $d$, the fan whose rays are generated by $e_1,\dots ,e_d,-{\sum }_i{e}_i$ and whose cones are generated by proper subsets of those rays is complete. It is the fan of projective space ${\mathbb{P}}^d$.