On terminal Fano 3-folds with 2-torus action (Q2800180)

The main result of this paper is the classification of terminal \(\mathbb{Q}\)-factorial Fano 3-folds of Picard number one with an effective two-torus action by explicitly listing their Cox rings. By a terminal Fano variety, we mean a complex normal projective variety with at worst terminal singularities. Although the complete classification of smooth Fano 3-folds is known by \textit{V. A. Iskovskikh} [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 516--562 (1977; Zbl 0363.14010); Izv. Akad. Nauk SSSR, Ser. Mat. 42, 506--549 (1978; Zbl 0407.14016)] and \textit{S. Mori} and \textit{S. Mukai} [Adv. Stud. Pure Math. 1, 101--129 (1983; Zbl 0537.14026)], the singular case is still unknown. Note that if a terminal Fano variety admits a torus action of complexity one, then the Cox ring has an explicit description as a complete intersection. To such a Fano variety, the authors associate the \textit{anticanonical complex}, which gives characterizations of singularities as in the toric case. By analyzing anticanonical complexes and using a software package made by Hausen-Keicher, the authors finally obtain the main result.