The classification of Fano 3-folds with torus embeddings
Let X be a smooth projective 3-fold over an algebraically closed field k. X is called a Fano 3-fold if the anti-canonical divisor \(-K_ X\) of X is ample. Recently, \textit{V. A. Ishkovskikh} has developed the theory of Fano 3-folds [Math. USSR, Izv. 11, 485-527 (1977), respectively 12, 469-506 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 516-562 (1977; Zbl 0363.14010), respectively 42, 506-549 (1978; Zbl 0407.14016)] and has determined the structure of Fano 3-folds with Picard number 1. In this paper, we will consider Fano 3-folds with torus embedding (which we will call toric Fano 3-folds) and determine all the toric Fano 3-folds up to isomorphism. As result, we see that there are 18 toric 3-folds and their Picard numbers are at most 5.
scientific article; zbMATH DE number 3929171