Related papers: An integral version of Zariski decompositions on n…

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic $p > 5$. As a consequence, we deduce the Grauert-Riemenschneider vanishing theorem for excellent divisorial log terminal…

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 note, we observe several properties of arithmetic divisors on the projective line over Z and give their Zariski decompositions.

In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a…

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of…

We establish a generic vanishing theorem for surfaces in characteristic $p$ that lift to $W_2(k)$ and use it for surface classification of surfaces of general type with Euler characteristic 1 and large Albanese dimension.

We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their…

In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that…

We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments…

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

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality…

Let $\alpha$ be a big class on a compact K\"ahler manifold. We prove that a decomposition $\alpha=\alpha_1+\alpha_2$ into the sum of a modified nef class $\alpha_1$ and a pseudoeffective class $\alpha_2$ is the divisorial Zariski…

We obtain an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple which makes an ample line bundle D very ample. The proof for…

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…

We show that the anti-canonical bundle of any $\mathbb Q$-factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with…

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…

We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting…

As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel…

We prove that a standard realization of the direct image complex via the so-called Douady-Barlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of…

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.…