Timeline for Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
Current License: CC BY-SA 2.5
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 31, 2010 at 15:57 | answer | added | roy smith | timeline score: 3 | |
| Oct 30, 2010 at 21:26 | comment | added | Emerton | ... to be an introductory text (unlike Hartshorne, say). | |
| Oct 30, 2010 at 21:25 | comment | added | Emerton | Dear Jim, Your recollection of Weil's book is correct. It also spends a lot of time explaining (without using functorial language) that $k[x_1,\ldots,x_n]$ is the free object on $n$ generators in the category of $k$-algebras, and no time explaining the intuition underlying intersection theory. I think it's fair to regard Weil's book as being written for an audience who already understands geometry, with the goal of introducing a rigorous algebraic foundation for the subject they already know (which in particular would work in char. p). In particular, I don't think it was ever intended ... | |
| Oct 30, 2010 at 20:01 | answer | added | roy smith | timeline score: 21 | |
| Oct 21, 2010 at 22:43 | comment | added | Jim Humphreys | The question is valuable for bringing out the range of current views on the subject. Having first encountered algebraic geometry in a year-long course based on Weil's Foundations (taught by Tamagawa), I feel disoriented for life and can therefore sympathize with the dilemma faced by others who want to learn the subject in the "right" way but while making some contact with actual geometry. Weil's book, as I recall, has no pictures whatsoever. | |
| Oct 21, 2010 at 13:21 | comment | added | Allen Knutson | As I commented here, there are at least three ways that schemes generalize varieties, and it's needless and unfortunate that they are all taught at once. mathoverflow.net/questions/28496/… | |
| Oct 21, 2010 at 11:30 | answer | added | Sheikraisinrollbank | timeline score: 8 | |
| Oct 21, 2010 at 8:52 | answer | added | Angelo | timeline score: 21 | |
| Oct 21, 2010 at 4:59 | answer | added | Emerton | timeline score: 6 | |
| Oct 21, 2010 at 3:28 | comment | added | Harry Gindi | That is, sheaves are in some sense the universal way to organize local data attached to a topological space. For example, this is true when we look at holomorphic manifolds, where the important sheaf is the sheaf of holomorphic functions back to $\mathbf{C}$. The classical theory of varieties is an abstraction of that idea to the algebraic setting. | |
| Oct 21, 2010 at 3:22 | comment | added | Harry Gindi | Just a comment, but Ravi Vakil is teaching a full-year algebraic geometry course where he is doing a "schemes-first" approach. The great thing about the notes is that they are written so the reader does not have to check Matsumura every time an important fact from commutative algebra comes up. He's putting the notes online bi-weekly, so you may want to check those out. Regarding the question: There are still modern ways to cover classical algebraic geometry. It turns out that sheaf theory is a much better way to describe even classical varieties than the more classical ad-hoc methods. | |
| Oct 21, 2010 at 2:46 | history | asked | Daniel Barter | CC BY-SA 2.5 |