On the existence of almost Fano threefolds with del Pezzo fibrations

Abstract

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.

Exotic $$G_2$$-manifolds

“…The quadric must be smooth, for if Q is singular then it contains planes. The section of by such a plane would be a plane cubic, defining a class with v 2 = 0 and d = 3, which is ruled out by (7.5) (see Fukuoka [17,Lemma 2.4]). Now consider the class E := 2H −G, which has degree 5 and E 2 = 0.…”

Refinement of the classification of weak Fano threefolds with sextic del Pezzo fibrations

“…In the articles [14,15], Jahnke-Peternell-Radloff classified the possibilities of the anti-canonical degree and the Sarkisov links of such threefolds. Moreover, for each of the possibilities, Jahnke-Peternell-Radloff [15] and the author [10] found examples for that possibility.…”

“…For each possibility, it is natural to ask how to find examples. After the above works, the articles [4,7,10] treated such problems. Especially, the classification of almost Fano del Pezzo fibrations of degree d = 6 was completed by [28], [14], and the author's paper [10].…”

“…After the above works, the articles [4,7,10] treated such problems. Especially, the classification of almost Fano del Pezzo fibrations of degree d = 6 was completed by [28], [14], and the author's paper [10]. Moreover, such threefolds with terminal anti-canonical models were classified up to deformation by Takeuchi [28].…”

“…The existence of an example of No.10 (resp. No.14) was treated by the author in [10]. However, the author realized that the proof of the existence of examples of No.10 and No.14 in [10] has a gap by Joseph Cutrone's pointing out.…”