Synopses & Reviews
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.
Synopsis
Exploring fresh developments as well as fundamental concepts, Syzygies and Hilbert Functions presents highlights, conjectures, unsolved problems, and examples of Hilbert functions and resolutions. The book studies the role of these functions in the areas of algebraic geometry and combinatorics. It also outlines the current status of both the LPP and multiplicity conjectures. In addition, the book discusses bigraded rings, multigraded rings, and toric rings, which have received a lot of attention in recent research.