Fano threefolds of large Fano index and large degree

Abstract

Abstract. We classify Q-Fano threefolds of Fano index > 2 and big degree.

ℚ-Fano threefolds of index 7

“…Throughout this paper the ground field k is supposed to be algebraically closed of characteristic 0. We use the notation of the papers [Pro07], [Pro10], [Pro13]. In particular, B(X) is the basket of singularities of a terminal threefold X. .…”

“…Thus there are 10 numerical "candidate varieties", i.e. collections of numerical invariants (see [B + ], [Pro10,Prop. 3.6]).…”

“…Among them, one case (with q Q (X) = 10) is not realized geometrically [Pro10] and the the remaining nine cases examples [BS07] are known. Furthermore, in five cases, the corresponding Q-Fano threefolds are completely described [Pro10], [Pro13]. The situation becomes more complicated for larger values of This work is supported by the Russian Science Foundation under grant 14-50-00005.…”

K-semistable Fano manifolds with the smallest alpha invariant

“…But in either case, −K X ∼ 8A where A is an effective divisor, which implies that α(X) ≤ 1 8 since (X, A) is not klt, a contradiction. Now assume that qQ(X) = 4, by [Pro13,Lemma 8.3], Cl(X) is torsionfree and qW(X) = qQ(X), hence there is a Weil divisor A such that −K X ∼ 4A. If g(X) ≥ 22, then by [Pro13, Theorem 1.2(vi)], X ≃ P 3 or X 4 ⊂ P (1, 1, 1, 2, 3).…”

“…Moreover, by classification of Q-Fano 3-fold with Q-factorial terminal singularities and ρ(X) = 1 with large Fano index due to Prokhorov [Pro10,Pro13], we prove the following: Theorem 1.5. Let X be a K-semistable Q-Fano 3-fold with Q-factorial terminal singularities and ρ(X) = 1.…”