Kaori Suzuki

This paper considers Q-Fano 3-folds X with ρ = 1. The aim is to determine the maximal Fano index f of X. We prove that f ≤ 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space P(3, 4, 5, 7). We also consider all possibility of X for f ≥ 9. 0. Introduction We say that X is a Q-Fano variety if it has only terminal singularities, the anticanonical Weil divisor −KX is ample, and it satisfies Q-factorial. X is Q-factorial if for an arbitrary Weil divisor D on X, there exists a positive integer r such that rD is a Cartier divisor. For suchX, there are two indices, the Gorenstein index, the smallest positive integer r for rKX is Cartier, and the Fano index f , which is the largest positive integer such that −KX = fA for a Weil divisor A. In this paper, we shall prove that Q-Fano index f ≤ 19, and determine the case of Q-Fano 3-folds X of Picard number ρ(X) = 1 over C with f ≥ 11. The Main Theorem is the following: Theorem 0.1. Let X be a Q-Fano 3-fold over C which is Q-factorial and satisfies ρ(X) = 1. Let f(X) be the Q-Fano index of X. Then 1. max f(X) = 19, and f(P(3, 4, 5, 7)) = 19. 2. If f(X) = 19, then the Hilbert series of X equals that of P(3, 4, 5, 7), namely 1/(1 − t)(1− t)(1 − t)(1− t). 3. f ∈ {1, . . . , 10, 11, 13, 17, 19} The classification of smooth Fano varieties has already studied by Fano and Iskovskikh [I], so we are only interested in the singular case. Sano [Sa] studied a similar problem in the case the Gorenstein index is less than or equal to the Fano index. Borisov and Borisov [BB] studied toric Fano 3-folds. Acknowledgements. The author would like to express her deep gratitude to Professor Miles Reid for his valuable comments and unceasing encouragement. She also thanks Professor Youichi Miyaoka, Professor Keiji Oguiso and Dr. Hiromichi Takagi for the discussion. Special thanks to Dr. Gavin Brown for teaching how to program in Magma. This paper was written during the author’s stay at Univ. of Warwick, U.K., 2001. 1