On the dual of the mobile cone
Let \(X\) be a normal projective complex variety. There are several cones one can define inside the vector spaces \(N^1(X)_{\mathbb R}\) and \(N_1(X)_{\mathbb R}\), and it is one of the basic questions which ones are dual to each other with respect to the intersection pairing. For instance, it is classical that the nef cone is dual to the cone of effective curves, and it is the main result of the paper [\textit{S. Boucksom, J.-P. Demailly, M. Păun} and \textit{T. Peternell}, ``The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension'', \url{arXiv:math/0405285}] that the pseudoeffective cone is dual to the cone of movable curves. The paper under review considers the question which cone is dual to the cone of movable divisors; and it gives a partial answer in the spirit of [\textit{C. Araujo}, Math. Z. 264, No. 1, 179--193 (2010; Zbl 1189.14006)] and [\textit{B. Lehmann}, J. Algebr. Geom. 21, No. 3, 473--493 (2012; Zbl 1253.14014)]. The author defines the cone of curves which are birationally movable in codimension \(1\) (b-mov\(^1\) curves), and proves that it is dual (in a certain region) to the movable cone; the precise statement is Theorem 1.1 in the paper. In particular, if \(X\) is a Fano variety, it is shown that the movable cone is dual to the cone of b-mov\(^1\) curves.
scientific article; zbMATH DE number 6097117
scientific article