Cohomology of Vector Bundles and Syzygies PDF Download
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Author: Jerzy Weyman
Publisher: Cambridge University Press
ISBN: 9780521621977
Category : Mathematics
Languages : en
Pages : 404
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Book Description
The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.
Author: Jerzy Weyman
Publisher: Cambridge University Press
ISBN: 9780521621977
Category : Mathematics
Languages : en
Pages : 404
Get Book Here
Book Description
The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.
Author: Marian Aprodu
Publisher: American Mathematical Soc.
ISBN: 0821849646
Category : Mathematics
Languages : en
Pages : 138
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Book Description
The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.
Author: E. Graham Evans
Publisher: Cambridge University Press
ISBN: 0521314119
Category : Mathematics
Languages : en
Pages : 137
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Book Description
This 1985 book covers from first principles the theory of Syzygies.
Author: J. Le Potier
Publisher: Cambridge University Press
ISBN: 9780521481823
Category : Mathematics
Languages : en
Pages : 260
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Book Description
This work consists of two sections on the moduli spaces of vector bundles. The first part tackles the classification of vector bundles on algebraic curves. The author also discusses the construction and elementary properties of the moduli spaces of stable bundles. In particular Le Potier constructs HilbertSHGrothendieck schemes of vector bundles, and treats Mumford's geometric invariant theory. The second part centers on the structure of the moduli space of semistable sheaves on the projective plane. The author sketches existence conditions for sheaves of given rank, and Chern class and construction ideas in the general context of projective algebraic surfaces. Professor Le Potier provides a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.
Author: Jonathan M. Fraser
Publisher: Cambridge University Press
ISBN: 1108478654
Category : Mathematics
Languages : en
Pages : 287
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Book Description
The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.
Author: Luchezar L. Avramov
Publisher: Cambridge University Press
ISBN: 0521831954
Category : Mathematics
Languages : en
Pages : 7
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Book Description
This book describes the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology, and combinatorics.
Author: M. P. Brodmann
Publisher: Cambridge University Press
ISBN: 0521513634
Category : Mathematics
Languages : en
Pages : 514
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Book Description
On its original publication, this algebraic introduction to Grothendieck's local cohomology theory was the first book devoted solely to the topic and it has since become the standard reference for graduate students. This second edition has been thoroughly revised and updated to incorporate recent developments in the field.
Author: Alexander Polishchuk
Publisher: Cambridge University Press
ISBN: 0521808049
Category : Mathematics
Languages : en
Pages : 308
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Book Description
Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.
Author:
Publisher: American Mathematical Soc.
ISBN: 0821883763
Category :
Languages : en
Pages : 190
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Book Description
Author: David Eisenbud
Publisher: Cambridge University Press
ISBN: 1107017084
Category : Mathematics
Languages : en
Pages : 633
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Book Description
3264, the mathematical solution to a question concerning geometric figures.
