Syzygies and Hilbert Functions

Syzygies and Hilbert Functions

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ISBN 9781584888604
Cat# C8601

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Features

  • Presents highlights, conjectures, unsolved problems, and examples of Hilbert functions and resolutions
  • Covers topics at the interface of commutative algebra, algebraic geometry, and combinatorics
  • Discusses the important invariant of Castelnuovo-Mumford regularity
  • Surveys two challenging conjectures: the LPP conjecture and the multiplicity conjecture
  • Describes bigraded rings, multigraded rings, and toric rings
  • Summary

    Hilbert functions and resolutions are both central objects in commutative algebra and fruitful tools in the fields of algebraic geometry, combinatorics, commutative algebra, and computational algebra. Spurred by recent research in this area, Syzygies and Hilbert Functions explores fresh developments in the field as well as fundamental concepts.

    Written by international mathematics authorities, the book first examines the invariant of Castelnuovo-Mumford regularity, blowup algebras, and bigraded rings. It then outlines the current status of two challenging conjectures: the lex-plus-power (LPP) conjecture and the multiplicity conjecture. After reviewing results of the geometry of Hilbert functions, the book considers minimal free resolutions of integral subschemes and of equidimensional Cohen-Macaulay subschemes of small degree. It also discusses relations to subspace arrangements and the properties of the infinite graded minimal free resolution of the ground field over a projective toric ring. The volume closes with an introduction to multigraded Hilbert functions, mixed multiplicities, and joint reductions.

    By surveying exciting topics of vibrant current research, Syzygies and Hilbert Functions stimulates further study in this hot area of mathematical activity.

    Table of Contents

    Some Results and Questions on Castelnuovo-Mumford
    Regularity
    Marc Chardin

    Hilbert Coefficients of Ideals with a View toward Blowup Algebras Alberto Corso and Claudia Polini

    A Case Study in Bigraded Commutative Algebra
    David Cox, Alicia Dickenstein and Hal Schenck

    Lex-Plus-Powers Ideals
    Christopher A. Francisco and Benjamin P. Richert

    Multiplicity Conjectures
    Christopher A. Francisco and Hema Srinivasan

    The Geometry of Hilbert Functions
    Juan C. Migliore

    Minimal Free Resolutions of Projective Subschemes of Small Degree
    Uwe Nagel

    Infinite Free Resolutions over Toric Rings
    Irena Peeva

    Resolutions and Subspace Arrangements
    Jessica Sidman

    Multigraded Hilbert Functions and Mixed Multiplicities
    Irena Swanson

    Index