A type of the Lefschetz hyperplane section theorem on $${\mathbb{Q}\,}$$ -Fano 3-folds with Picard number one and $${\frac{1}{2}(1,1,1)}$$ -singularities

Abstract

We prove a type of the Lefschetz hyperplane section theorem on \({\mathbb{Q}\,}\) -Fano 3-folds with Picard number one and \({\frac{1}{2}(1,1,1)}\) -singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi–Yau 3-fold X with Picard number one whose invariants are $$\left(H_X^3,\, c_2 (X) \cdot H_X, \,{{e}} (X) \right) = (8, 44, -88),$$ where HX, e(X) and c2(X) are an ample generator of Pic(X), the topological Euler characteristic number and the second Chern class of X respectively.