Deformations of Weak Fano 3-folds with Only Terminal Singularities

In this article, we prove that any Q-factorial weak Fano 3-fold with only terminal singularities has a smoothing. 0. Introduction Definition 0.1. Let X be a normal Gorenstein projective variety of dimension 3 over C which has only terminal singularities. (1) If −KX is ample, we call X a Fano 3-fold. (2) If −KX is nef and big, we call X a weak Fano 3-fold. Definition 0.2. Let X be a normal Gorenstein projective variety of dimension 3 with only terminal singularities. Let (∆, 0) be a germ of the 1-parameter unit disk. Let f : X → (∆, 0) be a small deformation of X over (∆, 0). We call f a smoothing of X when the fiber Xs = f (s) is smooth for each s ∈ (∆, 0) \ {0}. We treat the following problem in this paper: Problem. Let X be a weak Fano 3-fold with only terminal singularities. When X has a smoothing ? For the case of Fano 3-fold X , X has a smoothing by the result of Namikawa and Mukai ([Na 3],[Mukai]). Moreover by the method of Namikawa, we can show the following theorem: Theorem 0.3. (Namikawa, Takagi) (Cf. [Na 3],[T]) Let X be a weak Fano 3-fold with only terminal singularities. Assume that there exists a birational projective morphism π : X → X̄ from X to a Fano 3-fold with only canonical singularities X̄ such that dim(π(x)) ≤ 1. Then X has a smoothing. In this paper, we will show the following theorem: Main Theorem. Let X be a weak Fano 3-fold with only terminal singularities. (1) The Kuranishi space Def(X) of X is smooth. (2) There exists f : X → (∆, 0) a small deformation of X over (∆, 0) such that the fiber Xs = f (s) has only ordinary double points for any s ∈ (∆, 0)\{o}. (3) If X is Q-factorial, then X has a smoothing. We remark that if the condition of (3) “Q-factorial” is dropped, then there is an example that X remains singular under any small deformation [Cf. (3,7)].