Some remarks on cones of partially ample divisors (Q2810577)

Let \(X\) be a smooth complex projective variety of dimension \(n\). A line bundle \(L\) on \(X\) is said to be \(q\)-ample if for every coherent sheaf \(\mathcal{F}\) there exists \(m_0 \in \mathbb{Z}\) such that \(h^i(X, \mathcal{F} \otimes L^{\otimes m})=0\) for all \(m>m_0\) and \(i>q\). Observe that \(0\)-ampleness is ampleness and it can be proved that \(q\)-amplenes is a notion of numerical nature. Hence, it extends to the real space \(N^1(X)\) of numerical classes of divisors providing a series of cones: NEWLINE\[NEWLINE\mathrm{Amp}(X)=0\mathrm{Amp}(X) \subset 1\mathrm{Amp}(X) \subset \dots \subset n\mathrm{Amp}(X)=N^1(X)NEWLINE\]NEWLINE The goal of the paper under review is to (partially) describe these cones when \(0<q<n-1\). To be precise fix \(q=1\) and consider \(K_X\) the class of the canonical divisor. The \(K_X\)-visible part of the boundary of the nef cone is defined as the set of classes of divisors \(D\) such that the line \(K_X D\) meets \(\mathrm{Nef}(X)\) in the single point \(D\).NEWLINENEWLINEThe first result (see Theorem 19, first Theorem in the Introduction, for a precise statement) shows that, with two exceptions (blow-ups in codimension \(2\) subvarieties and conic bundles) \(\mathrm{Amp}(X)\) and \(1\mathrm{Amp}(X)\) look the same when observed from \(K_X\). The more negative \(K_X\) the more powerful this result, being the limit case when \(X\) is Fano or unless not zero and \(-K_X\) nef. In this case (with the same exceptions, see Cor. 25) the equality \(\mathrm{Amp}(X)=1\mathrm{Amp}(X)\) holds. Similar results are provided for \(q>1\), see Theorem 31.