Related papers: Big arithmetic divisors on the projective spaces o…

We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give…

We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor, we give formulae for its arithmetic…

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…

We characterize all projective K3 surfaces on which every integral pseudoeffective divisor admits an integral Zariski decomposition, using an explicit, terminating finite-step algorithm.

We give an example that the volume of an $\mathbb{R}$-divisor on a family of complex smooth surfaces jumps at infinite many prime divisors in the base. Our example follows the construction in \cite{1}.

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after…

Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is…

Any ample Cartier divisor D on a projective variety X is strictly nef (i.e. D.C>0 for any effective curve C on X). In general, the converse statement does not hold. But this is conjectured to be true for anticanonical divisors. The present…

Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'',…

We discuss different generalizations of Zariski decomposition, relations between them and connections with finite generation of divisorial algebras.

In this paper, we give a numerical characterization of nef arithmetic R-Cartier divisors of C^0-type on an arithmetic surface. Namely an arithmetic R-Cartier divisor D of C^0-type is nef if and only if D is pseudo-effective and deg(D^2) =…

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n…

For a compact hyperk\"ahler manifold X, we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a (holomorphic) Lagrangian fibration. We also prove that any…

Let $D$ be a big integral divisor on a smooth projective surface $X$. In this paper, we study Noether-type inequalities for $D$. The key ingredient is the introduction of a numerical invariant $\mathfrak{e}(D)$, which depends only on the…

We prove that, on a smooth threefold, pseudoeffective divisors with closed and one-dimensional diminished base locus have birationally a Fujita-Zariski decomposition.

In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic…

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the…

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].

In this paper, we prove that given a flat generically smooth morphism between smooth projective varieties with $F$-pure closed fibers, if the source space is Fano, weak Fano or a variety with the nef anti-canonical divisor, then so is the…

Let $X\subset\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$,…