Jumping of the nef cone for Fano varieties

Among all projective algebraic varieties, Fano varieties (those with ample anticanonical bundle) can be considered the simplest. Birkar, Cascini, Hacon and McKernan showed that the Cox ring of a Fano variety, the ring of all sections of all line bundles, is finitely generated [4]. This implies a fundamental fact about the birational geometry of a Fano variety: there are only finitely many small Q-factorial modifications of the variety, parametrized by a chamber decomposition of the movable cone into rational polyhedral cones (the nef cones of the modifications). See Kollár-Mori [21] or Hu-Keel [14] for definitions. There are also strong results about deformations of Fano varieties. For any deformation of a Q-factorial terminal Fano variety X0, de Fernex and Hacon showed that the Cox ring deforms in a flat family [5, Theorem 1.1, Proposition 6.4]. This had been proved for smooth Fanos by Siu [29, Corollary 1.2]. In other words, all line bundles have the “same number” of sections on X0 as on deformations of X0. It follows that the movable cone remains constant when X0 is deformed [5, Theorem 6.8]. To be clear, we use “deformation” to mean a nearby deformation; for example, the statement on the movable cone means that the movable cone is constant for t in some open neighborhood of 0, for any flat family X → T with fiber X0 over a point 0 ∈ T . When a Q-factorial terminal Fano variety X0 is deformed, de Fernex and Hacon asked whether the chamber decomposition of the movable cone also remains constant [5, Remark 6.2]. This would say in particular that the nef cone, or dually the cone of curves, remains constant under deformations of X0. The answer is positive in dimension at most 3, and also in dimension 4 when X0 is Gorenstein [5, Theorem 6.9]. In any dimension, Wísniewski showed that the nef cone of a smooth Fano variety remains constant under deformations [30, 31]. In this paper, we show that the blow-up of P along a line degenerates to a Q-factorial terminal Fano 4-fold X0 with a strictly smaller nef cone (Theorem 2.1). Therefore the results by de Fernex and Hacon on deformations of 3-dimensional Fanos are best possible. The example is based on the existence of high-dimensional flips which deform to isomorphisms, generalizing the Mukai flop. This phenomenon will be common, and we give a family of examples in various dimensions, including a Gorenstein example in dimension 5 (Theorem 1.1). The examples also disprove the “volume criterion for ampleness” on Q-factorial terminal Fano varieties [5, Question 5.5]. In view of Wísniewski’s theorem, it would be interesting to describe the largest class of Fano varieties for which the nef cone remains constant under deformations. It will not be enough to assume that the variety is Q-factorial and terminal, but the optimal assumption should also apply to many Fanos which are not Q-factorial or terminal.