Does anybody knows about good overview on intersection theory. The book of Fulton has very hard language. Does there exist simple overview on this topic with many examples?

5$\begingroup$ Have you looked at his book "Introduction to Intersection Theory in Algebraic Geometry"? I recall that being gentler. $\endgroup$– Dylan MorelandJan 20, 2011 at 18:30

1$\begingroup$ Ravi Vakil's intersection theory notes are also good.. $\endgroup$– J.C. OttemJan 20, 2011 at 18:40

4$\begingroup$ ..but if it's a survey you are after, I'd suggest appendix C of Hartshorne. $\endgroup$– J.C. OttemJan 20, 2011 at 18:41
3 Answers
Dear Klim, when you say "the" book, i suppose you mean Intersection Theory published by Springer . However Fulton has written a much more elementary overview called Introduction to Intersection Theory in Algebraic Geometry, published by the AMS in their Regional Conference Series in Mathematics , Number 54, which is only 74 page long and quite friendly.
There is also a great survey of Intersection Theory by Joël Riou here and Archibald's Master Thesis on Intersection Theory for surfaces there.

$\begingroup$ Dear Georges, when one is new to intersection theory, would you recommend reading Fulton's introduction book before starting with "the" book by Fulton? Or can one start right away, provided the necessary knowledge of scheme theory is already present? $\endgroup$– JoachimDec 12, 2013 at 10:22

1$\begingroup$ Dear @Joachim, I think "the" book by Fulton is incredibly difficult because of its terseness, even if one has a good knowledge of scheme theory. Personally I have resigned myself to use it as a reference and have given up the hope of reading it from cover to cover, and many colleagues have adopted a similar attitude. I think Eisenbud and Harris's book mentioned by Arnav in his answer is much more suitable as a textbook. $\endgroup$ Dec 12, 2013 at 12:51

1$\begingroup$ The link to Joël Riou's survey is now broken. Anybody can point to a working link? $\endgroup$– QfwfqMay 6, 2016 at 14:13
Eisenbud and Harris are coming out with a book on intersection theory, "3264 and all that", and if you know Harris's style at all, you'll know it's chock full of downtoearth examples that should be right along the lines of what you're looking for. (Sorry to recommend a book that's not strictly speaking published yet, but it does sound like exactly what you're asking!)

$\begingroup$ Just downloaded it. This reference is great. Thanks. $\endgroup$– JoëlFeb 24, 2011 at 3:11
This is certainly not an overview of intersection theory as a whole, but for its classical roots I highly recommend:
Steven L. Kleiman, Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445–482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.
I found this article to be both completely lucid and completely fascinating  and I am someone who, in general, has no great interest in intersection theory or (especially) Schubert calculus.