Algebraic Geometry and Arithmetic CurvesThis 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.
01. Algebraic geometry - Sheaves (Nickolas Rollick)
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. Reine Angew. Crelle's Journal , Robinson Award for best article in the Can.
This book provides a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic.
at the heart of the sea book
Math 8320 -- Algebraic Curves: MWF 1:25-2:15, 326 Boyd
Note : This is a study group and not a TCC lecture course. For a list of TCC course in Autumn click here. This study group will be trying to understand various topics in Algebraic Geometry and how they apply in a Number Theory context, with hopefully some concrete examples. All are welcomed and no pre-requisite in Algebraic Geometry are required although having a general knowledge of varieties will help. We will meet on Thursday between 9 and 11, from the 10th of October to 19th of December. The plan is to have a presentation for an hour and use the second hour as a discussion. Please contact me if you want to be on the mailing list organising the topics to cover.
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.