Related papers: On effective non-vanishing and base-point-freeness

We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective…

The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is…

Let $X$ be a horospherical $G$-variety and let $D$ be an effective $\mathbb{Q}$-divisor of $X$ that is stable under the action of a Borel subgroup $B$ of $G$ and such that $D+K\_X$ is $\mathbb{Q}$-Cartier. We prove, using Bott-Samelson…

Let $F(X,Y)=Y^d+a_1(X)Y^{d-1}+...+a_d(X)$ be a polynomial in $n+1$ variables $(X,Y)=(X_1,...,X_n,Y)$ with coefficients in an algebraically closed field K. Assuming that the discriminant $D(X)=\disc_Y F(X,Y)$ is nonzero we investigate the…

Let $A$ be a Banach algebra admitting a bounded approximate unit and satisfying property $\mathbb{B}$. Suppose $T: A \rightarrow X$ is a continuous linear map, where $X$ is an essential Banach $A$-bimodule. We prove that the following…

We consider the operators $H_0=M_0^{-1}(x) P(D)$ and $H =M^{-1} (x) P(D)$ where $M_0 (x)$ and $M (x)$ are positively definite bounded matrix-valued functions and $P(D)$ is an elliptic differential operator. Our main result is that the wave…

For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that…

In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on…

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria…

Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.

We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral.…

Let $\epsilon, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of…

Let $\overline{\mathrm{Mov}}^k(X)$ be the closure of the cone $\mathrm{Mov}^k(X)$ generated by classes of effective divisors on a projective variety $X$ with stable base locus of codimension at least $k+1$. We propose a generalized version…

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…

Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n\geq 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $|\mathcal{L}|$ contains an element…

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a…

We give a negative answer to a question of Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa, and Testa on effective divisors of positive self-intersection on smooth projective surfaces. The main result of this paper is obtained by…

We extend the Bass-Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is…

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we give some concrete examples and properties of strongly…

Let $X'$ be a complex projective manifold, $\dim X'>1$, $Z$ a connected analytic subset of codimension one which is the support of a nef effective Cartier divisor $D$ on $X'$, $X:=X'\setminus Z$. Let $\kappa(D)$ be the Iitaka dimension of…