On the existence of a weak Zariski decomposition on projectivized vector bundles
type: Item
instance of: scholarly article
title: On the existence of a weak Zariski decomposition on projectivized vector bundles
zbMATH Open document ID: 1329.14024
author: Roberto Muñoz
author: Fulvio Di Sciullo
author: Luis E. Solá Conde
published in: Geometriae Dedicata
publication date: 2015-12-16T00:00:00Z
review text: 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 mai…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 6521463
zbMATH Keywords: Zariski decompositions
zbMATH Keywords: projective bundles
zbMATH Keywords: stability of vector bundles
zbMATH Keywords: positivity
MaRDI profile type: Publication
full work available at URL: s10711-015-0082-8
OpenAlex ID: W2181426456
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DOI: 10.1007/S10711-015-0082-8
description: scientific article; zbMATH DE number 6521463