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Author Liu, Qing
Title Algebraic Geometry and Arithmetic Curves
Imprint Oxford : Oxford University Press, Incorporated, 2002
©2006
book jacket
Descript 1 online resource (594 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Oxford Graduate Texts in Mathematics (0-19-961947-6) Ser. ; v.6
Oxford Graduate Texts in Mathematics (0-19-961947-6) Ser
Note Intro -- Contents -- 1 Some topics in commutative algebra -- 1.1 Tensor products -- 1.1.1 Tensor product of modules -- 1.1.2 Right-exactness of the tensor product -- 1.1.3 Tensor product of algebras -- 1.2 Flatness -- 1.2.1 Left-exactness: flatness -- 1.2.2 Local nature of flatness -- 1.2.3 Faithful flatness -- 1.3 Formal completion -- 1.3.1 Inverse limits and completions -- 1.3.2 The Artin-Rees lemma and applications -- 1.3.3 The case of Noetherian local rings -- 2 General properties of schemes -- 2.1 Spectrum of a ring -- 2.1.1 Zariski topology -- 2.1.2 Algebraic sets -- 2.2 Ringed topological spaces -- 2.2.1 Sheaves -- 2.2.2 Ringed topological spaces -- 2.3 Schemes -- 2.3.1 Definition of schemes and examples -- 2.3.2 Morphisms of schemes -- 2.3.3 Projective schemes -- 2.3.4 Noetherian schemes, algebraic varieties -- 2.4 Reduced schemes and integral schemes -- 2.4.1 Reduced schemes -- 2.4.2 Irreducible components -- 2.4.3 Integral schemes -- 2.5 Dimension -- 2.5.1 Dimension of schemes -- 2.5.2 The case of Noetherian schemes -- 2.5.3 Dimension of algebraic varieties -- 3 Morphisms and base change -- 3.1 The technique of base change -- 3.1.1 Fibered product -- 3.1.2 Base change -- 3.2 Applications to algebraic varieties -- 3.2.1 Morphisms of finite type -- 3.2.2 Algebraic varieties and extension of the base field -- 3.2.3 Points with values in an extension of the base field -- 3.2.4 Frobenius -- 3.3 Some global properties of morphisms -- 3.3.1 Separated morphisms -- 3.3.2 Proper morphisms -- 3.3.3 Projective morphisms -- 4 Some local properties -- 4.1 Normal schemes -- 4.1.1 Normal schemes and extensions of regular functions -- 4.1.2 Normalization -- 4.2 Regular schemes -- 4.2.1 Tangent space to a scheme -- 4.2.2 Regular schemes and the Jacobian criterion -- 4.3 Flat morphisms and smooth morphisms -- 4.3.1 Flat morphisms -- 4.3.2 Étale morphisms
4.3.3 Smooth morphisms -- 4.4 Zariski's 'Main Theorem' and applications -- 5 Coherent sheaves and Cech cohomology -- 5.1 Coherent sheaves on a scheme -- 5.1.1 Sheaves of modules -- 5.1.2 Quasi-coherent sheaves on an affine scheme -- 5.1.3 Coherent sheaves -- 5.1.4 Quasi-coherent sheaves on a projective scheme -- 5.2 Cech cohomology -- 5.2.1 Differential modules and cohomology with values in a sheaf -- 5.2.2 Cech cohomology on a separated scheme -- 5.2.3 Higher direct image and flat base change -- 5.3 Cohomology of projective schemes -- 5.3.1 Direct image theorem -- 5.3.2 Connectedness principle -- 5.3.3 Cohomology of the fibers -- 6 Sheaves of differentials -- 6.1 Kähler differentials -- 6.1.1 Modules of relative differential forms -- 6.1.2 Sheaves of relative differentials (of degree 1) -- 6.2 Differential study of smooth morphisms -- 6.2.1 Smoothness criteria -- 6.2.2 Local structure and lifting of sections -- 6.3 Local complete intersection -- 6.3.1 Regular immersions -- 6.3.2 Local complete intersections -- 6.4 Duality theory -- 6.4.1 Determinant -- 6.4.2 Canonical sheaf -- 6.4.3 Grothendieck duality -- 7 Divisors and applications to curves -- 7.1 Cartier divisors -- 7.1.1 Meromorphic functions -- 7.1.2 Cartier divisors -- 7.1.3 Inverse image of Cartier divisors -- 7.2 Weil divisors -- 7.2.1 Cycles of codimension 1 -- 7.2.2 Van der Waerden's purity theorem -- 7.3 Riemann-Roch theorem -- 7.3.1 Degree of a divisor -- 7.3.2 Riemann-Roch for projective curves -- 7.4 Algebraic curves -- 7.4.1 Classification of curves of small genus -- 7.4.2 Hurwitz formula -- 7.4.3 Hyperelliptic curves -- 7.4.4 Group schemes and Picard varieties -- 7.5 Singular curves, structure of Pic[sup(0)](X) -- 8 Birational geometry of surfaces -- 8.1 Blowing-ups -- 8.1.1 Definition and elementary properties -- 8.1.2 Universal property of blowing-up
8.1.3 Blowing-ups and birational morphisms -- 8.1.4 Normalization of curves by blowing-up points -- 8.2 Excellent schemes -- 8.2.1 Universally catenary schemes and the dimension formula -- 8.2.2 Cohen-Macaulay rings -- 8.2.3 Excellent schemes -- 8.3 Fibered surfaces -- 8.3.1 Properties of the fibers -- 8.3.2 Valuations and birational classes of fibered surfaces -- 8.3.3 Contraction -- 8.3.4 Desingularization -- 9 Regular surfaces -- 9.1 Intersection theory on a regular surface -- 9.1.1 Local intersection -- 9.1.2 Intersection on a fibered surface -- 9.1.3 Intersection with a horizontal divisor, adjunction formula -- 9.2 Intersection and morphisms -- 9.2.1 Factorization theorem -- 9.2.2 Projection formula -- 9.2.3 Birational morphisms and Picard groups -- 9.2.4 Embedded resolutions -- 9.3 Minimal surfaces -- 9.3.1 Exceptional divisors and Castelnuovo's criterion -- 9.3.2 Relatively minimal surfaces -- 9.3.3 Existence of the minimal regular model -- 9.3.4 Minimal desingularization and minimal embedded resolution -- 9.4 Applications to contraction -- canonical model -- 9.4.1 Artin's contractability criterion -- 9.4.2 Determination of the tangent spaces -- 9.4.3 Canonical models -- 9.4.4 Weierstrass models and regular models of elliptic curves -- 10 Reduction of algebraic curves -- 10.1 Models and reductions -- 10.1.1 Models of algebraic curves -- 10.1.2 Reduction -- 10.1.3 Reduction map -- 10.1.4 Graphs -- 10.2 Reduction of elliptic curves -- 10.2.1 Reduction of the minimal regular model -- 10.2.2 Néron models of elliptic curves -- 10.2.3 Potential semi-stable reduction -- 10.3 Stable reduction of algebraic curves -- 10.3.1 Stable curves -- 10.3.2 Stable reduction -- 10.3.3 Some sufficient conditions for the existence of the stable model -- 10.4 Deligne-Mumford theorem -- 10.4.1 Simplifications on the base scheme -- 10.4.2 Proof of Artin-Winters
10.4.3 Examples of computations of the potential stable reduction -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z
This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. It explains both theory and applications clearly and includes essential background methods. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book is suitable for graduate students
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Liu, Qing Algebraic Geometry and Arithmetic Curves Oxford : Oxford University Press, Incorporated,c2002 9780198502845
Subject Curves, Algebraic.;Arithmetical algebraic geometry
Electronic books
Alt Author Erne, Reinie
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