Jun 162014

刚买了一本 David Mumford 的选集 “Selected Papers: On Algebraic Geometry, Including Correspondence with Grothendieck”.

Selected Papers On Algebraic Geometry, Including Correspondence with Grothendieck

Selected Papers: On Algebraic Geometry, Including Correspondence with Grothendieck

这是 David Mumford 选集的第二卷. 第一卷是 2004 年的 “Selected Papers: On the Classification of Varieties and Moduli Spaces”.

这个第二卷非常吸引人的地方, 是第二部分的 57 封信. 这些信的时间跨度大概是 25 年. 除了第一封 Grothendieck 1958 年 8 月 5 日写给Zariski, 余下的 56 封信都是 David Mumford 与 Grothendieck 的往来. 这些信绝大部分发生在 1960-1970 年间, 只有 5 封是 1984-1986 年间写成. 奇怪的是, 这些信, 有 51 封是 Grothendieck 所写, David Mumford 给 Grothendieck 的信只有 6 封.

本书是 Springer 在 2010 年推出, 并不久远.  让人费解的是, 不知道什么原因, 在 Springer 的网站找不到电子版.

Oct 292013

真心佩服 Springer! 出版的好书无数: 无数的系列, 每个系列都是几十, 几百. 很多的资讯都很独家, 极具价值!

代数几何最”浅”的书, 大概是 Vladimir I. Arnold 的 “Real Algebraic Geometry“!  Springer 刚刚出来英译本. 六章加一个附录, 刚好 100 页! 本书是面向高中生的讲座. 不过, 不懂一点拓扑学, 微积分, 射影几何, 是不可能完全看懂的!

虽然代数几何有不同的切入路径, 但是想入门代数几何, 最起码要在掌握基本的抽象代数之后, 最好能有较强的射影几何(Projective Geometry)基础.

不建议 David Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms, 以及 Harris, Algebraic Geometry: A First Course. 从这样的书, 不会学到多少东西, 尽管这些书都很容易读, 要求的预备知识也很少.

根据很多人的看法, Bertrametti, Carletti, Gallarati, Bragadin, Lectures on Curves, Surfaces and Projective Varieties 很精彩.

1. Daniel Bump, Algebraic Geometry.

The prerequisites for the textbook are fairly minimal. Although it does discuss commutative algebra, there is a flavor of geometry pervasive throughout the entire text.

2. Holme, A Royal Road to Algebraic Geometry.

3. Shafarevich, Basic Algebraic Geometry, vol. 1, 2

英文译本第三版, 出来没多久.

4. Perrin Algebraic Geometry an Introduction.

5. Miles Reid, undergraduate algebraic geometry.

Sep 182013

Steve pointed out the thing that makes EGA difficult to read is not that it is dense, but rather that it is gigantic.

Robin Hartshorne’s book algebraic geometry is an edulcorated version of Grothendieck and Dieudonné’s EGA, which changed algebraic geometry forever.

EGA was so notoriously difficult that essentially nobody outside of Grothendieck’s first circle(roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .

Things began to change with the appearance of Mumford’s mimeographed notes in the 1960’s, the celebrated Red Book, which allowed the man in the street(well, at least the streets near Harvard) to be introduced to scheme theory.

Then, in 1977, Hartshorne’s revolutionary textbook  algebraic geometry was published. With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.

However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.

GTM 52 的精华是第 2, 3章, 分别介绍 Scheme 和它上面的 Cohomollogy theory.

GTM 52 有习题 464 道. 这本书的习题, 非常重要! 当然, 习题也不一定必须一个一个全部做完.

Jun 202013
Basic Algebraic Geometry 1

Basic Algebraic Geometry 1

The third Edition of  “Basic Algebraic Geometry” has just been published.

Shafarevich’s Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, “For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.”

Shafarevich’s book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.

The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann–Roch theorem for curves, including a proof from first principles.

Basic Algebraic Geometry 2

Basic Algebraic Geometry 2

The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the “Shafarevich conjecture”.

The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.