A cone theorem for nef curves (Q2894542)

The paper under review gives a structure theorem for the cone of nef curves \(\overline{NM}_{1}(X)\) on a dlt pair \((X,\Delta)\). \(\overline{NM}_{1}(X)\) is defined to be the cone dual to the cone of pseudo-effective divisors under the intersection product.NEWLINENEWLINEThe main theorem of the paper says that an enlargement of the cone of nef curves can be described along the same lines of the Cone Theorem. More precisely, there are countably many \((K_{X}+\Delta)\)-negative movable curves \(C_{i}\) such that NEWLINE\[NEWLINE\overline{NE}_{1}(X)_{K_{X}+\Delta\geq 0}+\overline{NM}_{1}(X)=\overline{NE}_{1}(X)_{K_{X}+\Delta\geq 0}+\overline{\sum\mathbb{R}_{\geq 0}[C_{i}]},NEWLINE\]NEWLINE where \(\overline{NE}_{1}(X)\) is the cone of effective curves. A slightly weaker statement was proved in [\textit{C. Araujo}, Math. Z. 264, No. 1, 179--193 (2010; Zbl 1189.14006)]. Furthermore, the author describes an example where the enlargement is necessary.NEWLINENEWLINEThe above theorem is part of a conjecture due to Batyrev. He conjectured also that the rays \(\mathbb{R}_{\geq 0}[C_{i}]\) only accumulate along the hyperplane \((K_{X}+\Delta)^{\bot}\). In the last section of the paper under review, the author proves that if \(\Delta\) is big then Batyrev's conjecture follows from termination of flips.