On classification of ℚ-fano 3-folds of Gorenstein index 2. I

We formulate a generalization of K. Takeuchi’s method to classify smooth Fano 3-folds and use it to give a list of numerical possibilities of ℚ-Fano 3-foldsXwith PicX= ℤ(−2KX) andh0(−KX) ≥ 4 containing index 2 pointsPsuch that (X, P) ≃ ({xy+z2+ua= 0}/ℤ2(1, 1, 1, 0),o) for somea∈ ℕ. In particular we prove that then (–KX)3≤ 15 andh0(–KX) ≤ 10. Moreover we show that such anXis birational to a simpler Mori fiber space.