# Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics (0-19-961947-6))

## Qing Liu

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Book by Liu Qing

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Will be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field. ( EMS)

This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. ( Zentralblatt MATH)

A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. ( Zentralblatt MATH)

Reseña del editor:

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.

The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group.

The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford.

The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

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## 1.Algebraic Geometry and Arithmetic Curves

Editorial: OUP Oxford (2016)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción OUP Oxford, 2016. Paperback. Estado de conservación: New. PRINT ON DEMAND Book; New; Publication Year 2016; Not Signed; Fast Shipping from the UK. No. book. Nº de ref. de la librería ria9780198502845_lsuk

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## 2.Algebraic Geometry and Arithmetic Curves

Editorial: Oxford University Press (2002)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, 2002. HRD. Estado de conservación: New. New Book. Delivered from our US warehouse in 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND.Established seller since 2000. Nº de ref. de la librería IP-9780198502845

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## 3.Algebraic Geometry and Arithmetic Curves

Editorial: Oxford University Press (2002)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, 2002. HRD. Estado de conservación: New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Nº de ref. de la librería I1-9780198502845

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## 4.Algebraic Geometry and Arithmetic Curves (Hardback)

Editorial: Oxford University Press, United Kingdom (2002)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, United Kingdom, 2002. Hardback. Estado de conservación: New. 236 x 156 mm. Language: English . Brand New Book ***** Print on Demand *****. 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. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski s Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck s duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo s criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises. Nº de ref. de la librería APC9780198502845

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## 5.Algebraic Geometry and Arithmetic Curves (Hardback)

Editorial: Oxford University Press, United Kingdom (2002)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, United Kingdom, 2002. Hardback. Estado de conservación: New. 236 x 156 mm. Language: English . Brand New Book ***** Print on Demand *****.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. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski s Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck s duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo s criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises. Nº de ref. de la librería APC9780198502845

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## 6.Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics)

Editorial: Oxford University Press (2017)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, 2017. Hardcover. Estado de conservación: New. This item is printed on demand. Nº de ref. de la librería 0198502842

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## 7.Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics)

Editorial: Oxford University Press (2017)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, 2017. Hardcover. Estado de conservación: New. This item is printed on demand. Nº de ref. de la librería P110198502842

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## 8.Algebraic Geometry And Arithmetic Curves

ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Estado de conservación: New. This item is Print on Demand - Depending on your location, this item may ship from the US or UK. Nº de ref. de la librería POD_9780198502845

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## 9.Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics)

Editorial: Oxford University Press, USA (2002)
ISBN 10: 0198502842 ISBN 13: 9780198502845
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Descripción Oxford University Press, USA, 2002. Hardcover. Estado de conservación: New. 1st. Nº de ref. de la librería DADAX0198502842

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## 10.Algebraic Geometry and Arithmetic Curves

Editorial: OUP Oxford
ISBN 10: 0198502842 ISBN 13: 9780198502845
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