When are Zariski chambers numerically determined? (Q339422)

The authors study a question whether the so-called Zariski chambers are numerically determined by simple Weyl chambers. A celebrated result due to Zariski (improved by Fujita) claims that for every \(\mathbb{R}\)-pseudoeffective divisor \(D\) on a smooth projective surface \(X\), there exist effective \(\mathbb{R}\)-divisors \(P_{D}\) and \(N_{D}\) such that NEWLINE\[NEWLINED = P_{D} + N_{D},NEWLINE\]NEWLINE where \(P_{D}\) is nef, \(N_{D}\) is either empty or \(N_{D} = \sum_{i=1}^{s}a_{i}N_{i}\) with \(a_{i} > 0\), and the intersection matrix of irreducible components \([C_{i}.C_{j}]_{s\times s}\) is negative definite, and for every \(i \in \{1,\dots,s\}\) one has \(P_{D}.C_{i}=0\). In [J. Reine Angew. Math. 576, 209--233 (2004; Zbl 1055.14007)] \textit{T. Bauer} et al. gave a fundamental result providing a characterization of the big cone \(\text{Big}(X)\) in the language of \textit{Zariski chambers}. Suppose that \(P\) is big and nef divisor, then we define the Zariski chamber \(\Sigma_{P}\) associated to \(P\) as NEWLINE\[NEWLINE\begin{aligned} & \Sigma_{P}:= \{B \in \text{Big}(X) : \text{ irreducible components of } N_{B} \text{ are the irreducible curves on } X \\ &\text{ that intersect } P \text{ with multiplicity } 0 \}. \end{aligned}NEWLINE\]NEWLINE It can be shown that Zariski chambers yield a locally finite decomposition of the big cone \(\text{Big}(X)\) into locally polyhedral subcones such that the support of the negative part of the Zariski decomposition of all divisors in the subcone is constant. Let \(\mathcal{N}(X)\) be the set of all irreducible negative curves on \(X\), i.e., curves \(C\subset X\) with \(C^{2} < 0\). Each curve \(C \in \mathcal{N}(X)\) one defines the hyperplane in \(\text{NS}_{\mathbb{R}}(X)\) of \(X\) by NEWLINE\[NEWLINEC^{\perp} = \{D : D.C = 0\} \subset \text{NS}_{\mathbb{R}}(X),NEWLINE\]NEWLINE and the decomposition of the set NEWLINE\[NEWLINE\text{Big}(X) \setminus \bigcup_{C \in \mathcal{N}(X)}C^{\perp}NEWLINE\]NEWLINE into connected components yields a decomposition of (an open and dense subset of) the cone \(\text{Big}(X)\) into subcones -- these components are called simple Weyl chambers of \(X\).NEWLINENEWLINEThe natural idea is to compare these two decompositions and, hopefully, find some relations between them. Since Zariski chambers are in general neither open nor closed, and by definition Weyl chambers are open, thus the natural approach to this problem is to focus on interiors of Zariski chambers. The main result of the paper is the following criterion.NEWLINENEWLINETheorem 1. Let \(X\) be a smooth projective surface. The following conditions are equivalent:NEWLINENEWLINEi) the interior of each Zariski chamber on \(X\) is a simple Weyl chamber,NEWLINENEWLINEii) if two irreducible negative curves \(C_{1} \neq C_{2}\) on \(X\) meet, then NEWLINE\[NEWLINEC_{1}.C_{2} \geq \sqrt{ C_{1}^{2} . C_{2}^{2}}.NEWLINE\]NEWLINENEWLINENEWLINEIn other words, the second condition tells us that the support of the non-trivial negative part of the Zariski decomposition of every big divisor on \(X\) consists of pairwise disjoint curves.NEWLINENEWLINEThe above result is inspired by the earlier result due to \textit{T. Bauer} and \textit{M. Funke} for \(K3\) surfaces [Forum Math. 24, No. 3, 609--625 (2012; Zbl 1242.14007)]NEWLINENEWLINEIn the second part, the authors focus on Enriques surfaces. Another result provides a link between chamber decompositions and elliptic fibrations.NEWLINENEWLINETheorem 2. Let \(X\) be an Enriques surface. Then the following conditions are equivalent:NEWLINENEWLINEi) the simple Weyl chambers and the interiors of Zariski chambers on \(X\),NEWLINENEWLINEii) every fiber of every elliptic fibration on \(X\) has at most two components.