Browsing by Author "Szemberg, Tomasz"

Now showing 1 - 6 of 6
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    Arrangements of lines
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, Tomasz
    We discuss certain open problems in the context of arrangements of lines in the plane.
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    Mini-Workshop: Arrangements of Subvarieties, and their Applications in Algebraic Geometry
    (Zürich : EMS Publ. House, 2016) Di Rocco, Sandra; Harbourne, Brian; Szemberg, Tomasz
    While arrangements of hyperplanes have been studied in algebra, combinatorics and geometry for a long time, recent discoveries suggest that they (and more generally arrangements of nonlinear subvarieties) play an even more fundamental role in major problems in algebraic geometry than has yet been understood. The workshop brought into contact experts from commutative algebra and algebraic geometry working on these problems – it provided opportunities to get updated on the latest developments through talks of the participants, but also reserved time for working groups in which participants brainstormed ideas and insights in the context of high-intensity discussions aimed at initiating immediate progress on proposed problems, thereby setting the stage for on-going collaborations after the workshop.
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    Mini-Workshop: Linear Series on Algebraic Varieties
    (Zürich : EMS Publ. House, 2010) Di Rocco, Sandra; Harbourne, Brian; Szemberg, Tomasz
    Linear series have long played a central role in algebraic geometry. In recent years, starting with seminal papers by Demailly and Ein-Lazarsfeld, local properties of linear series – in particular local positivity, as measured by Seshadri constants – have come into focus. Interestingly, in their multi-point version they are closely related to the famous Nagata conjecture on plane curves. While a number of important basic results are available by now, there are still a large number of open questions and even completely open lines of research.
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    Mini-Workshop: Negative Curves on Algebraic Surfaces
    (Zürich : EMS Publ. House, 2014) Küronya, Alex; Müller-Stach, Stefan; Szemberg, Tomasz
    Negative curves play a prominent role in the geometry of projective surfaces. They occur naturally as the irreducible components of exceptional loci of resolutions of surface singularities, at the same time, they are closely related to the geometry of the effective cone, and thus form an important building block of the Minimal Model Program. In the case of surfaces, classes of negative curves span extremal rays of the Mori cone. Any knowledge about them on a given surface reveals important information on linear series as well.
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    On the containment problem
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Szemberg, Tomasz; Szpond, Justyna
    Mathematicians routinely speak two languages: the language of geometry and the language of algebra. When translating between these languages, curves and lines become sets of polynomials called “ideals”. Often there are several possible translations. Then the mystery is how these possible translations relate to each other. We present how geometry itself gives insights into this question.
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    Unexpected Properties of the Klein Configuration of 60 Points in P³
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2020) Pokora, Piotr; Szemberg, Tomasz; Szpond, Justyna
    Felix Klein in course of his study of the regular and its symmetries encountered a highly symmetric configuration of 60 points in P³. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the 60 reflection planes in the group G₃₁ in the Shephard-Todd list. In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities 4; 2 and 2. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in P³ with the surprising property that their general projection to P² is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in P³ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.