Torus factor

A torus factor of a toric variety is a product factor ${\mathbb{G}}_m^r$ in a toric decomposition $X\cong Y\times {\mathbb{G}}_m^r$. A nontrivial factor has $r>0$. This is stronger than merely containing a dense torus, which every toric variety does.

For a toric fan $\Sigma$ in a lattice $N$, let $N^{\prime }=N\cap {\mathrm{span}}_{\mathbb{R}}({\bigcup }_{\sigma \in \Sigma }\sigma )$. It is a saturated sublattice. Choosing a complementary lattice $N^{\prime \prime }$ givesThe complementary splitting need not be canonical. Thus a proper linear span of the whole fan supplies a nontrivial torus factor.

For the fan consisting of one coordinate ray and zero in ${\mathbb{Z}}^2$, the affine coordinate ring is $k[x,y,y^{-1}]$, giving ${\mathbb{A}}^1\times {\mathbb{G}}_m$. This variety has no torus-fixed point because the second multiplicative coordinate cannot be fixed by every scaling. A fan with only lower-dimensional cones can still span the whole lattice, so absence of full-dimensional individual cones alone does not imply a torus factor.