Geodesic
Elementary Birational Maps between Mori Toric Fiber 3-Spaces
Matematičeskie zametki, Tome 78 (2005) no. 1, pp. 132-139

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In the present paper, we classify elementary equivariant birational maps (links) between Mori toric fiber 3-spaces. These links are naturally divided into several classes, depending on the dimension of the bases. For classes containing finitely many links, we present a complete list, in other cases we provide a local description (see the statements in Sec. 4). Almost all the proofs are of combinatorial nature, which is why we present proofs of the results from Sec. 3 only. 
@article{MZM_2005_78_1_a12,
     author = {K. A. Shramov},
     title = {Elementary {Birational} {Maps} between {Mori} {Toric} {Fiber} {3-Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {132--139},
     publisher = {mathdoc},
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     number = {1},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_1_a12/}
}
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K. A. Shramov. Elementary Birational Maps between Mori Toric Fiber 3-Spaces. Matematičeskie zametki, Tome 78 (2005) no. 1, pp. 132-139. http://geodesic.mathdoc.fr/item/MZM_2005_78_1_a12/