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I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.

What I am looking for are references for classical enumerative geometry, back before Hilbert's 15th Problem asked people to actually make it work as rigorous mathematics. Are there good references for the original (flawed!) arguments? I'd prefer perhaps something more recent than the original papers and books (many are hard to find, and even when I can, I tend to be a bit uncomfortable just handling 150 year old books if there's another option.)

More specifically, are there modern expositions of the original arguments by Schubert, Zeuthen and their contemporaries? And if not, are there translations or modern (20th century, say...) reprints of their work available, or are scanned copies available online (I couldn't find much, though I admit my German is awful enough that I might have missed them by not having the right search terms, so I'm hoping for English review papers or the like, though I'll deal with it if I need to.)

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I wish you (sincerely!) good luck with this. Some famous mathematician (Rota?) once lamented that there are too few people who do the work of reading old mathematics, putting it context and reinterpreting it for a contemporary audience, because this requires skill and interest in both mathematics and history. Most historians of mathematics do not have the mathematical skills, and most mathematicians know and care much more about contemporary work (especially their own) than past mathematics, even if this causes each generation to independently rediscover many results in its own language. –  Pete L. Clark Jan 8 '10 at 6:17
Thanks, I've had a hobby in history of math since I was an undergrad and I think that it's something I'd like to pursue as a sideline. Specifically, enumerative geometry has interested me, but I found out last May that I just didn't have the stomach for the kinds of arguments going on in GW theory, and switched subfields. Probably partly the influence of Anders Buch, one of my ugrad profs. And the big reason I'm asking is that I feel kind of weird holding an actual copy of Schubert's "Kalkul der Abzahlenden Geometrie" in my hands without damaging it. –  Charles Siegel Jan 8 '10 at 6:40

5 Answers 5

Have you looked at Semple and Roth, Introduction to Algebraic Geometry? It was published in 1949 and contains a wealth of classical results (there is a chapter devoted to enumerative geometry). Going back a bit further, both German and French Encyclopaedias of Mathematical Sciences published in the early 20th century had surveys of algebraic geometry. Moving in the opposite direction, Fulton's "Intersection theory" discusses applications of his theory to classical enumerative geometry problems where excessive intersections play crucial role (such as finding the number of conics touching 5 given ones).

I know you said you've decided to move away from GW theory, but I thought I'd just throw it in here: Sheldon Katz's book "Enumerative geometry and string theory" (Student Mathematical Library, vol 32) is actually very readable.

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You could also take a look at Schubert Calculus by Kleiman and Laksov in the Monthly, Vol. 79, No. 10, pp 1061-1082 or the monograph Geometry of Coxeter Groups by H. Hiller.

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I've seen Kleiman/Laksov, though not the Hiller monograph, but generally seems rather more modern than I was looking for, though thanks for pointing out the Hiller. –  Charles Siegel May 28 '10 at 1:54
up vote 3 down vote accepted

And actually, as a partial answer to my own question, I just stumbled across Schubert's "Kalkul" on Google Books, and it looks complete, which makes me rather happy, though other portions of the question still stand.

EDIT: A friend of mine has informed me that Zeuthen's "Lehrbuch" is also there, and now it's linked.

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Yeah, it's complete, you can download the pdf, and it even appears to have a good bibliography, index, and endnotes. –  Jonas Meyer Jan 8 '10 at 6:54

Here is a link to a more than 3 page list of works on enumerative geometry from the second half of the 19th century, including those of Schubert, Zeuthen, and many others. (Perhaps it will be a useful guide for searching for scanned copies of the originals, e.g. through UPenn fulltext subscriptions.)

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Thanks, I'll definitely be sorting through some of that. –  Charles Siegel Jan 8 '10 at 6:41

I do know of one article taking a historical approach to Schubert calculus:

Kleiman, Steven L. 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 am much less of a Schubertist than the average Berkeley/Harvard-educated mathematician with research interests in algebraic geometry, but nevertheless I found this article to be fascinating reading.

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