Qing Liu Algebraic Geometry And Arithmetic Curves Pdf

qing liu algebraic geometry and arithmetic curves pdf

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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.

Ravi Vakil's Publications and Preprints, etc.

When R is the ring of integers Z , this intuition depends on the prime ideal spectrum Spec Z being seen as analogous to a line. In even more generality, arithmetic surfaces can be defined over Dedekind schemes, a typical example of which is the spectrum of the ring of integers of a number field which is the case above. An arithmetic surface is then a regular fibered surface over a Dedekind scheme of dimension one. Arithmetic surfaces over Dedekind domains are the arithmetic analogue of fibered surfaces over algebraic curves. In higher dimensions one may also consider arithmetic schemes.

What is this course about? However, if it so happens that the polynomials have their coefficients in a smaller field that is not algebraically closed such as the field of rational numbers, then it makes sense and there may be good reason to ask for solutions with coefficients in that field. But this is often a subtle issue which usually involves Galois theory, even when the field is that of the real numbers and this explains why it was not a good idea to start out that way. Things becomes even more complicated if the algebraically closed field is replaced by a ring, for instance the ring of integers. Such questions are by no means uninteresting, as many natural questions in number theory can be stated that way. This foundational work was carried out during a relatively short period under the leadership of A. At the same time, experience has taught us that the scheme setting is ill-suited for a first acquaintance with algebraic geometry, and this is why most of this course is concerned with Algebraic Geometry over an algebraically closed field.

Algebraic Geometry and Arithmetic Curves

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Algebraic Geometry and Arithmetic Curves

Any comments, corrections or suggestions would be greatly appreciated. I haven't posted TeX files of articles with complicated figures. Kubo, Discrete Math. Thesis, Harvard University, , under the supervision of Joe Harris.

Offers end pm EST. Author: Kentaro Mitsui Journal: J. Algebraic Geom. Abstract: We study invariants of an elliptic fibration over a complete discrete valuation ring with algebraically closed residue field.

Algebraic Geometry and Arithmetic Curves

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Liu Published Mathematics.

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