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020    |z9780198502845 
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035    (Au-PeEL)EBL430442 
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050  4 QA565.L68 2006 
082 0  516.352 
100 1  Liu, Qing 
245 10 Algebraic Geometry and Arithmetic Curves 
264  1 Oxford :|bOxford University Press, Incorporated,|c2002 
264  4 |c©2006 
300    1 online resource (594 pages) 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
490 1  Oxford Graduate Texts in Mathematics (0-19-961947-6) Ser. 
       ;|vv.6 
505 0  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 
505 8  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 
505 8  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 
505 8  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 
520    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 
588    Description based on publisher supplied metadata and other
       sources 
590    Electronic reproduction. Ann Arbor, Michigan : ProQuest 
       Ebook Central, 2020. Available via World Wide Web. Access 
       may be limited to ProQuest Ebook Central affiliated 
       libraries 
650  0 Curves, Algebraic.;Arithmetical algebraic geometry 
655  4 Electronic books 
700 1  Erne, Reinie 
776 08 |iPrint version:|aLiu, Qing|tAlgebraic Geometry and 
       Arithmetic Curves|dOxford : Oxford University Press, 
       Incorporated,c2002|z9780198502845 
830  0 Oxford Graduate Texts in Mathematics (0-19-961947-6) Ser 
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