I am interested in blowups of normal toric varieties (i.e. toric varieties which come from a fan $\Sigma$). I know that blowups along torus-invariant closed subvarieties $V(\sigma), \sigma \in\Sigma$ are again normal toric varieties, whose fans can be constructed from the fan $\Sigma$. Also the blowup along a union of those torus-invariant closed subvarieties should be a normal toric variety, as we can realize this by a succession of those blowups. My question ist: Are there any other examples of subvarieties, where the blowup is again a normal toric variety or are (unions of) torus-invariant subvarieties the only subvarieties with this property? I am particularly interested in blowups along the image of toric morphisms - are these blowups always toric?
Thanks!