Geometry of Schemes
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Description: The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his co-workers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings' proof of the Mordell Conjecture).This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required.
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All the information you need in one place! Each Study Brief is a summary of one specific subject; facts, figures, and explanations to help you learn faster.
Copyright year: 2000
Publication date: 11/29/2001
Size: 6.25" wide x 9.50" long x 0.75" tall
|Schemes in General|
|The Functor of Points|
|Reduced Schemes over Algebraically Closed Fields|
|Reduced Schemes over Non-Algebraically Closed Fields|
|Attributes of Morphisms|
|Proj of a Graded Ring|
|Invariants of Projective Schemes|
|Flexes of Plane Curves|
|Singular Schemes and Discriminants|
|Double Point Loci|
|Schemes and Functors|
|The Functor of Points|
|Characterization of a Space by its Functor of Points|