On a generalized Batyrev's cone conjecture (Q2069627)

Let \((X, \Delta)\) be a klt pair. The Cone Theorem, a fundamental result in the Minimal Model Program, tells us that the Mori cone \(\overline{\mathrm{NE}}(X)\) decomposes into a \((K_X + \Delta)\)-nonnegative part and a countable sum of \((K_X + \Delta)\)-negative extremal rays satisfying some ``local finiteness'' condition. Another interesting cone, \(\overline{\mathrm{NM}}(X)\), is given by the closure of the cone spanned by \emph{movable} curves. A similar description, going back to Batyrev, can be given for the cone \(\overline{\mathrm{NM}}(X, \Delta) := \overline{\mathrm{NE}}(X)_{K_X+\Delta\ge0} + \overline{\mathrm{NM}}(X)\). The purpose of this paper is to treat both results in a unified way and find a common generalization. To this end, the authors consider curves \emph{movable in codimension \(\ell\)}, that is, curves which move in a family covering a subvariety of codimension at most \(\ell\). The closure of these curve classes defines the cone \(\overline{\mathrm{NM}}^\ell(X)\), but a birational version \(\overline{\mathrm{bNM}}^\ell(X)\) taking into account also small \(\mathbb Q\)-factorial modifications is more important. The main result (Theorem~1.3) is a cone theorem for \(\overline{\mathrm{NE}}^\ell(X, \Delta) := \overline{\mathrm{NE}}(X)_{K_X+\Delta\ge0} + \overline{\mathrm{bNM}}^\ell(X)\). The cases \(\ell = \dim X - 1\) and \(\ell = 0\) correspond to the aforementioned results.