Related papers: When are Zariski chambers numerically determined?
Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior…
The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study…
Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a…
Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. In the present paper we present an algorithm…
Inspired by the work of Bauer, K\"uronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support…
We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms…
We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the so far available method and allows therefore to treat surfaces of high Picard number, where huge…
We characterize all projective K3 surfaces on which every integral pseudoeffective divisor admits an integral Zariski decomposition, using an explicit, terminating finite-step algorithm.
The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on…
Let $X$ be a K3 surface defined over a number field $K$. Assume that $X$ admits a structure of an elliptic fibration or an infinite group of automorphisms. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational…
In this paper, we characterize smooth projective surfaces on which every integral pseudoeffective divisor has an integral Zariski decomposition.
We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108].
Let $X$ be a rational elliptic surface with elliptic fibration $\pi:X\to\Bbb{P}^1$ over an algebraically closed field $k$ of any characteristic. Given a conic bundle $\varphi:X\to\Bbb{P}^1$ we use numerical arguments to classify all…
This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces…
We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in…
The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a…
In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the…
Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. For irreducible symplectic manifolds Boucksom provided a characterization of his divisorial Zariski decomposition in terms of the…
The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the…
Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point…