Product fan

The product of fans ${\Sigma }_1\subseteq N_{1,\mathbb{R}}$ and ${\Sigma }_2\subseteq N_{2,\mathbb{R}}$ consists of the cones ${\sigma }_1\times {\sigma }_2$. Its toric variety is $X_{{\Sigma }_1}\times X_{{\Sigma }_2}$.