Geometry of the locus of polynomials of degree 4 with iterative roots

We study polynomial iterative roots of polynomials and describe the locus of complex polynomials of degree 4 admitting a polynomial iterative square root.Comment: 7 pages, accepted by Central European Journal of Mathematic

Bounds on Seshadri constants on surfaces with Picard number 1

In this note we improve a result of Steffens on the lower bound for Seshadri constants in very general points of a surface with one-dimensional N\'eron-Severi space. We also show a multi-point counterpart of such a lower bound.Comment: 7 pages, to appear in Comm. Algebr

An effective and sharp lower bound on Seshadri constants on surfaces with Picard number 1

We study lower bounds on Seshadri constants at arbitrary points on surfaces with Picard number 1.Comment: 8 pages, to appear in J. Algebr

On the containment problem

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.Comment: 13 pages, 1 figur

On the Seshadri constants of adjoint line bundles

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 bundles there are explicit lower bounds depending only on the dimension of the underlying variety. In the surface case, where the optimal lower bound is 1/2, we characterize all possible values in the range between 1/2 and 1 -- there are surprisingly few. As expected, one obtains even more restrictive results for the Seshadri constants of adjoints of very ample line bundles. Our description of the border case in this situation makes use of adjunction-theoretical results on surfaces. Finally, we study Seshadri constants of adjoint line bundles in the multi-point setting.Comment: Added Remark 3.3, which points out an improvement to the lower bound in Theorem 3.2 by using G. Heier's resul

Seshadri constants and the generation of jets

In this paper we explore the connection between Seshadri constants and the generation of jets. It is well-known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants

Remarks on the Nagata Conjecture

2000 Mathematics Subject Classification: 14C20, 14E25, 14J26.The famous Nagata Conjecture predicts the lowest degree of a plane curve passing with prescribed multiplicities through given points in general position. We explain how this conjecture extends naturally via multiple point Seshadri constants to ample line bundles on arbitrary surfaces. We show that if there exist curves of unpredictable low degree, then they must have equal multiplicities in all but possibly one of the given points. We use this restriction in order to obtain lower bounds on multiple point Seshadri constants on a surface. We discuss also briefly a seemingly new point of view on the Nagata Conjecture via the bigness of the involved linear series