000			02003 a2200181 4500
020			_a9780199202492 (pbk.)
082			_a516.35 LIU
100			_aLiu, Qing
245			_aAlgebraic Geometry and Arithmetic Curves / _cQing Liu
260			_bOUP, _aNew York: _cc2002.
300			_axv, 577p.
504			_aIncludes Bibliography (557- 561) & Index (562-577).
520			_aThis new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with 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 desingularization (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 fundamental theorem of stable reduction of Deligne-Mumford. This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students.
650			_aCurves, Algebraic
650			_aArithmetical algebraic geometry
650			_aGeometry, Algebraic
942			_cBK
999			_c2795 _d2795