00 6 on 3 - Folds of General Type with Χ = 1

There are many examples of 3-folds of general type with χ(O) = 1 found by Fletcher and Reid about twenty years ago. Iano-Fletcher has ever proved P12(X) ≥ 1 and P24(X) ≥ 2 for all minimal 3-folds X of general type with χ(OX) = 1. In this paper, we improve on Fletcher’s method. Our main results are: Pm ≥ 1 for all m ≥ 14, P18 ≥ 2, P20 ≥ 3 and that φm is birational onto its image for all m ≥ 88 which rewrites an earlier record of Kollár. A feature corollary is that P18(X) = 2 if and only if K X = 1 420 , pg(X) = 0 and X has virtue baskets Q of singularities: 2 of type 1 2 (1,−1, 1), 2 of type 1 3 (2,−2, 1), 1 of type 1 4 (3,−3, 1), 1 of type 1 5 (3,−3, 1) and 1 of type 1 7 (5,−5, 1). Under the above situation, P2(X) = P3(X) = P4(X) = · · · = P11(X) = 0, P12(X) = 1, P13(X) = 0, P14(X) = P15(X) = P16(X) = P17(X) = 1, P18(X) = P19(X) = 2, P20(X) = P21(X) = 3, P23(X) = 4, P24(X) = P25(X) = 5.