Related papers: Very general monomial valuations of $\mathbb{P}^2$…
T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata's 1959 open conjecture, which claims that the Seshadri constant of r>9 very general points of the projective plane is maximal. Here we prove that Nagata's…
We provide several equivalent conditions for a plane divisorial valuation of a smooth projective surface to be minimal with respect to an ample divisor. These conditions involve a valuative Seshadri constant and other global tools of the…
We provide a lower bound on the degree of curves of the projective plane $\mathbb{P}^2$ passing through the centers of a divisorial valuation $\nu$ of $\mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type…
We consider the last value $\hat{\mu} (\nu)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $\nu$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective…
Fujita's conjecture is known to be false in positive characteristic. We conjecture and give an approach to a new variant of Fujita's conjecture for the basepoint-freeness, very ampleness, and jet ampleness of linear systems of the form…
In the present paper we are concerned with the possible values of Seshadri constants. While in general every positive rational number appears as the local Seshadri constant of some ample line bundle, we point out that for adjoint line…
Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constant for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this…
The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.
One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics…
Let X_g=C^{(2)}_g be the second symmetric product of a very general curve of genus g. We reduce the problem of describing the ample cone on X_g to a problem involving the Seshadri constant of a point on X_{g-1}. Using this we recover a…
Motivated by a similar result of Dumnicki, K\"uronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface $X$ obtained by blowing up very general points of…
In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three…
We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the…
Let $\pi: X_r \rightarrow \mathbb P^2$ be a blow up of $\mathbb P^2$ at $r$ distinct points $p_1,p_2,\dots, p_r$. We study lower bounds for Seshadri constants of ample line bundles on $X_r$. First, we consider the case when the points lie…
In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of…
Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\varepsilon(L)$ of $L$ is defined as the infimum of Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is an interesting…
The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r $\ge$ 10 general points in the…
We study the Seshadri constants of cyclic coverings of smooth surfaces. The existence of an automorphism on these surfaces can be used to produce Seshadri exceptional curves. We give a bound for multiple Seshadri constants on cyclic…
In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants,…
Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional…