Abstract
If a non-Gorenstein Q \mathbb {Q} -Fano threefold with only cyclic quotient terminal singularities has anti-canonical Du Val K 3 K3 surfaces and the anti-canonical class generates the group of numerical equivalence classes of divisors, then the dimension of the space of global sections of the anti-canonical sheaf is shown to be not greater than ten. Such Q \mathbb {Q} -Fano threefolds with the dimension not less than nine are classified.
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