= Star subdivision
A <star subdivision> inserts the <toric ray> through a nonzero <toric lattice> vector $v$ lying in the support of a <toric fan>. For every <toric cone> containing $v$, replace it by the <toric cones> generated by $v$ together with its faces that do not contain $v$, retaining all resulting faces and all unaffected <toric cones>. The support is unchanged, and the induced <toric morphism> is proper and birational.
For the first quadrant in $\mathbb Z^2$, inserting $(1,1)$ splits it into the <toric cones> generated by $(1,0),(1,1)$ and by $(1,1),(0,1)$. Both determinants are one. This is the <toric fan> of the <blowup of affine space at the origin> for $\mathbb A^2$, with a new invariant exceptional <projective line>. General star subdivisions need not be blowups of smooth centers; the <toric blowup along an orbit closure> states the special smooth ray-sum case.<Toric Varieties by David A. Cox and others>[chapter3, star subdivisions and toric blowups]{ref}
Back to article page
