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3 results

  • Prolongations and Computational Algebra

  • Jessica Sidman, Seth Sullivant
  • Journal:
    Canadian Journal of Mathematics / Volume 61 / Issue 4 / 01 August 2009
    Published online by Cambridge University Press:
    20 November 2018, pp. 930-949
    Print publication:
    01 August 2009
    • Article
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  • We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

  • SUBSPACE ARRANGEMENTS DEFINED BY PRODUCTS OF LINEAR FORMS

  • ANDERS BJÖRNER, IRENA PEEVA, JESSICA SIDMAN
  • Journal:
    Journal of the London Mathematical Society / Volume 71 / Issue 2 / April 2005
    Published online by Cambridge University Press:
    06 April 2005, pp. 273-288
    Print publication:
    April 2005

  • The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3$ are also considered.

  • Lectures on the Geometry of Syzygies

  • Edited by Luchezar L. Avramov, University of Nebraska, Lincoln, Mark Green, University of California, Los Angeles, Craig Huneke, University of Kansas, Karen E. Smith, University of Michigan, Ann Arbor, Bernd Sturmfels, University of California, Berkeley
  • Mathematical Sciences Research Institute
  • Book:
    Trends in Commutative Algebra
    Published online:
    06 July 2010
    Print publication:
    13 December 2004, pp 115-152

  • Summary

    The theory of syzygies connects the qualitative study of algebraic varieties and commutative rings with the study of their defining equations. It started with Hilbert’s work on what we now call the Hilbert function and polynomial, and is important in our day in many new ways, from the high abstractions of derived equivalences to the explicit computations made possible by Gr¨obner bases. These lectures present some highlights of these interactions, with a focus on concrete invariants of syzygies that reflect basic invariants of algebraic sets.

    These notes illustrate a few of the ways in which properties of syzygies reflect qualitative geometric properties of algebraic varieties. Chapters 1, 3 and 4 were written by David Eisenbud, and closely follow his lectures at the introductory workshop. Chapter 2 was written by Jessica Sidman, from the lecture she gave enlarging on the themes of the first lecture and providing examples. The lectures may serve as an introduction to the book The Geometry of Syzygies [Eisenbud 1995]; in particular, the book contains proofs for the many of the unproved assertions here.