You'll have to study from other sources as well but I believe that this book does a pretty good job at motivating the abstract definitions. The theory of stacks originated with the work of Grothendieck and his students in the 60s. You should take a look at it and see if you like it, the book has a fair amount of examples and some exercises interspersed within the text also. The organizers are: Patrick Brosnan, Roy Joshua and Hsian-Hua Tseng. Note that the book that Arturo recommended is great also but it assumes you already know some algebraic geometry and its level is higher than Ueno's book. Artin, The implicit function theorem in algebraic geometry, Algebraic. It is not a short story of course, but again I prefer this type of approach at first, than having to deal with an unmotivated (and difficult) definition that strives for great generality but I have no idea of where it comes from and what is its purpose. This is a graduate level course in algebraic stacks on 7.5 credits. and then defines the necessary things in order to be able to define an affine scheme and a scheme (I mean, the concepts of a sheaf of rings, a ringed space, etc). Its strong points are the numerous, illuminating pictures. It contains serious material like Hilbert functions or blowing-up of ideals and calculations of just the right kind: neither trivial nor too technical. Then chapter two develops first some properties of this set of prime ideals, or prime spectrum of a ring, making it into a topological space with the Zariski topology. is an excellent survey of classical algebraic geometry at the intermediate level. In particular, it is noted how an extension of the definitions to include these cases would need to take into account not only the set of maximal ideals, but the set of all prime ideals. However, what I really like is that he motivates very carefully the passage from the definition of an affine algebraic variety as an irreducible algebraic set in an affine space $\mathbb(R), R )$, where $R$ is a finitely generated $k$ algebra.Īnd then at the end of the first chapter the author motivates the need for a more general theory, for example having in mind the needs of number theory, because since everything was done in the context of an algebraically closed field, then the arguments don't work for the fields (and rings) of interest in number theory. It does not go into cohomology and more advanced stuff, which is the subject of the other two books. So this first volume basically just develops the definitions of an affine scheme first and then of a scheme in general by "pasting" together affine schemes. Well, to be fair, this is only the first in a series of three books on the subject by the same author. Overall these will be aimed at helping you contribute efficiently to the Stacks project.įor those who are interested, we will also discuss how it is possible to contribute to the software that runs the Stacks project.I have found Kenji Ueno's book Algebraic Geometry 1: From Algebraic Varieties to Schemes to be quite satisfying in introducing the basic theory of schemes. Adding references to and finding mistakes in the Stacks project (and fixing them) as well as activities related to the use of LaTeX, Git, and GitHub. The Stacks project workshop will have some optional activities you won't see at other workshops. Note: If you are accepted to the workshop, we will later ask for your preferences for group/topic/mentor. This is all spelled out in Harris Algebraic Geometry, although not in the language of linear systems. You take any Q P, look at the line through P and Q, and see where it hits H. The workshop will be at the University of Michigan in Ann Arbor, from August 7 to August 11, 2023. The first example is projection from a point: pick a point P P n and a hyperplane H not containing the point. The list of currently confirmed mentors is: The sections on algebraic geometry in 'Mirror Symmetry' (Clay/AMS) are essentially a Crib Notes version of that paper and some of the classic CY and special geometry papers referred to above. Mathematicians not in the groups are welcome to come to these talks. There will also be some talks each day covering advanced topics in algebraic geometry. Part of this process will be seeing how one builds new theory from the foundations. Roughly the idea is that they will learn about and work on a single topic in a small group together with a mentor for a week. The participants who apply and are accepted will be graduate students, post-docs, or even senior researchers. Purely mathematical questions should NOT go here, instead, they belong on Math Stack Exchange. Questions tagged algebraic-geometry Ask Question Use for questions about algebraic geometry as it applies to physics. This will be a workshop in Algebraic Geometry, in topics (tangentially) related to the Stacks project. Newest 'algebraic-geometry' Questions - Physics Stack Exchange As of May 31, 2023, we have updated our Code of Conduct.
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