On the existence of a weak Zariski decomposition on projectivized vector bundles
In the present paper the authors study the concept of weak Zariski decompositions of pseudoeffective divisors on projectivized vector bundles. A classical result due to Fujita-Zariski tells us that on a smooth projective surface every pseudoeffective \(\mathbb{R}\)-divisor \(D\) can be uniquely expressed as \(D = P + N\), where \(P\) is nef, \(N\) is effective and satisfies that its irreducible components are orthogonal to \(P\) and, either \(N = 0\) or the intersection matrix of its components is negative definite. This fundamental result is a powerful tool in the theory of algebraic surfaces. In a similar manner many authors wanted to generalize the concept of Zariski decompositions to higher dimensional varieties and one of the most important approaches is due to \textit{N. Nakayama} [Zariski-decomposition and abundance. MSJ Memoirs 14. Tokyo: Mathematical Society of Japan (2004; Zbl 1061.14018)], where he constructed the so-called \(\sigma\)-decomposition. In the paper the authors study the concept of a \textit{weak Zariski decomposition}. Definition. Let \(D\) be a \(\mathbb{R}\)-divisor on a normal projective variety \(X\). We say that \(D\) has a \textit{weak Zariski decomposition} (WZD for short) if there exists a birational morphism \(f: W \rightarrow X\) from a normal variety \(W\), such that {\parindent=6mm \begin{itemize} \item[i)] \(f^* D = P+N\), where \(P, N\) are \(\mathbb{R}\)-divisors and the equality is numerical, \item [ii)] \(P\) is nef and \(N\) is effective. \end{itemize}} Moreover, if \(f: W \rightarrow X\) can be chosen as the identity, then we say that \(D\) admits a \textit{direct WZD}. It is worth pointing out that a WZD has important consequences, for instance the existence of WZD for divisors \(K_{X} + B\) is equivalent to the existence of a log minimal model for a log-canonical pair \((X,B)\) \textit{C. Birkar} [Math. Ann. 354, No. 2, 787--799 (2012; Zbl 1255.14016)], or it appears naturally in the context of Mori dream spaces. The main problem of the paper is the following. { Problem}. Consider a rank \(r\geq 2\) vector bundle \(\mathcal{E}\) on a smooth projective variety \(Z\) of Picard number one and let \(X = \mathbb{P}(\mathcal{E}) \rightarrow Z\) be its Grothendieck projectivization. Does a WZD (or a direct WZD) exist for any pseudoeffective divisor on \(X\)? The main result of the paper gives affirmative answers in several cases. Let us emphasize here that in the sequel notions of stable/unstable/semistable vector bundles are considered with respect to the slope stability. {Main Theorem}. Let \(\mathcal{E}\) be a vector bundle of rank \(r\) on a smooth projective variety \(Z\). {\parindent=6mm \begin{itemize} \item[1)] If \(Z\) is a curve, then for any pseudoeffective divisor on \(\mathbb{P}(\mathcal{E})\) \item [i)] a WZD exists and \item [ii)] a direct WZD exists if and only if either \(\mathcal{E}\) is semistable or \(\mathcal{E}\) is unstable and the pseudoeffective cone of its maximal destabilizing subbundle is closed. \item [2)] If \(Z\) is Fano of Picard number one, \(r=2\) and \(\mathcal{E}\) is normalized (i.e., \(c_{1}(\mathcal{E}) = 0,-1\)), then a direct WZD for any pseudoeffective on \(\mathbb{P}(\mathcal{E})\) exists when \item [i)] \(\mathcal{E}\) is unstable or semistable but not stable, and \item [ii)] \(\mathcal{E}\) is stable, the fourth Betti number of \(Z\) is equal to one, there exists a closed positive dimensional family of rational curves through the general point of \(Z\) and the first twist of \(\mathcal{E}\) having global sections is \(\mathcal{E}(1)\). \end{itemize}} The first case \(Z =\text{ curve }\) was investigated by \textit{N. Nakayama} [Section IV. 3, Zbl 1061.14018], but the authors present a new proof of this result using both technics related to the slope stability and some ideas around a characterization of the pseudoeffective cone of a projective bundle on a curve given by \textit{M. Fulger} [Math. Z. 269, No. 1--2, 449--459 (2011; Zbl 1230.14047)]. The proof of the second case is more involving. In Section 4 the authors provide some instructive examples.
scientific article; zbMATH DE number 6521463
scientific article