Presents highlights, conjectures, unsolved problems, and examples of Hilbert functions and resolutionsCovers topics at the interface of commutative algebra, algebraic geometry, and combinatoricsDiscusses the important invariant of Castelnuovo-Mumford regularitySurveys two challenging conjectures: the LPP conjecture and the multiplicity conjectureDescribes bigraded rings, multigraded rings, and toric rings
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
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
Christopher A. Francisco and Benjamin P. Richert
Christopher A. Francisco and Hema Srinivasan
The Geometry of Hilbert Functions
Juan C. Migliore
Minimal Free Resolutions of Projective Subschemes of Small Degree
Infinite Free Resolutions over Toric Rings
Resolutions and Subspace Arrangements
Multigraded Hilbert Functions and Mixed Multiplicities