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It was proved by J. A. Chen and M. Chen that a terminal Fano three-fold X satisfies . We show that a
-factorial terminal Fano three-fold X with
and
is a weighted hypersurface of degree 66 in
. By the same method, we also give characterizations for other 11 examples of weighted hypersurfaces of the form
in Iano-Fletcher’s list. Namely, we show that if a
-factorial terminal Fano three-fold X with
has the same numerical data as X6d, then X itself is a weighted hypersurface of the same type.
For a real number 
$0<\epsilon <1/3$, we show that the anti-canonical volume of an 
$\epsilon $-klt Fano 
$3$-fold is at most 
$3,200/\epsilon ^4$, and the order 
$O(1/\epsilon ^4)$ is sharp.
$\mathbb {P}(1,1,1,k)$ in torus-invariant subvarieties
This papers classifies toric Fano threefolds with singular locus 
$\{ \frac {1}{k}(1,1,1) \}$ for 
$k \in \mathbb {Z}_{\geq 1}$ building on the work of Batyrev (1981, Nauk SSSR Ser. Mat. 45, 704–717) and Watanabe–Watanabe (1982, Tokyo J. Math. 5, 37–48). This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore, we identify birational relationships between entries of the classification. For a fixed value 
$k \geq 4$, there are exactly two such toric Fano threefolds linked by a blowup in a torus-invariant line.
