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-folds X with Pic X = Z(-2K(x)) and h(0)(-K-X) greater than or equal to 4 containing index 2 points P such that (X, P) similar or equal to ({xy + z(2) + u(a) = 0}/Z(2)(1,1,1, 0), o) for some a is an element of N. In particular we prove that then (-K-X)(3) less than or equal to 15 and h(o)(-K-X) less than or equal to 10. Moreover we show that such an X is birational to a simpler Mori fiber space.