## what mistakes did the Italian algebraic geometers actually make?

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It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations (comm alg etc) were only just being laid, and possibly (as far as I know) due to the fact that in the 19th century not everyone had come round to the axiomatic way of doing things (perhaps in those days one could use geometric plausibility arguments and they would not be shouted down as non-rigorous and hence invalid? I have no real idea about how maths was done then).

But someone asked me for an explicit example of a false result "proved" by this school, and I was at a loss. Can anyone point me to an explicit example? Preferably a published paper that contained arguments which were at the time at least partially accepted by the community as being OK but in fact have holes in? Actually, to be honest I'd probably prefer some sort of English historical summary of such things, but I do have access to (living and rigorous) Italian algebraic geometers if necessary ;-)

EDIT: A few people have posted solutions which hang upon the Italian-ness or otherwise of the person making the mathemtical mistake. It was not my intention to bring the Italian-ness or otherwise of mathematicians into the question! Let me clarify the underlying issue: a friend of mine, interested in logic, asked me about (a) Grothendieck's point of view of set theory and (b) a precise way that one could formulate the statement that he "made algebraic geometry rigorous". My question stemmed from a desire to answer his.

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Are you aware of this question which has a similar flavor? mathoverflow.net/questions/17352/… – jc Mar 26 2010 at 13:39

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@jc: no ;-) Thanks! That question turned out to have a narrower remit I guess. – Kevin Buzzard Mar 26 2010 at 13:54

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ftp.mcs.anl.gov/pub/qed/archive/209 This illuminating email by David Mumford is a concise example of how a modern algebraic geometer might feel about the work of the Italian school. – bhwang Mar 26 2010 at 19:14

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Perhaps *the 19th century Italian school of algebraic geometry* should be the the 20th century... – Chandan Singh Dalawat Feb 11 2011 at 9:16

## 6 Answers

As for a result that was not simply incorrectly proved, but actually false, there is the case of the Severi bound(*) for the maximum number of singular double points of a surface in P^3. The prediction implies that there are no surfaces in P^3 of degree 6 with more than 52 nodes, but in fact there are such surfaces in P^3 with 64 nodes (and this is optimal).

(*) Francesco Severi; "Sul massimo numero di nodi di una superficie di dato ordine dello spazio ordinario o di una forma di un iperspazio." Ann. Mat. Pura Appl. (4) 25, (1946). 1--41.

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There is also Severi's incorrect proof of the irreducibility of the spaces of plane curves of degree d with r nodes. A proof of the result was given by Harris (ams.org/mathscinet/search/…). – damiano Mar 26 2010 at 16:20
I think I am forced to accept this answer as being precisely what I was looking for! However I find all the answers interesting: one thing this has going for it is that it has the most precise references in. Note to damiano: there exist people without access to mathscinet, and for them your answer might be rather more cryptic than it could be. – Kevin Buzzard Mar 27 2010 at 12:55
I have added an explicit reference to the paper by Severi. The reference for Harris is: Joe Harris; "On the Severi problem." Invent. Math. 84 (1986), no. 3, 445--461. – damiano Mar 28 2010 at 11:22

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[Added disclaimer: What follows is the product of probably faulty memory combined with a limited understanding in the first place, so should be taken with a grain of salt.]

Dear Kevin,

I believe that Brill--Noether of curves gives the kind of examples you are looking for. (My understanding, probably imperfect if not completely wrong, is that they made certain general position arguments about existence of linear systems that were just wrong, because they didn't realize that certain kinds of geometric condition were universal, and so, although they look special, are in fact general.)

You might try looking at the old papers of Harris (or maybe Eisenbud and Harris) about linear systems on curves.

Also, the introduction (by Zariski) to Zariski's collected works is interesting. He began in the Italian school, but then became instrumental in introducing algebraic tools.

Also, I think that the newest edition of his book on algebraic surfaces (a report on the results of the Italian school) has annotations by Mumford, which are very illuminating with regard to the differences and similarities between the Italian style and a more modern style.

P.S. Here's a way to imagine the kind of error one could make in general position arguments (although obviously any actual such error made by the Italians would be many times more subtle): Let $P_1,\ldots,P_8$ be eight points. Choose two elliptic curve $E_1$ and $E_2$ passing through the 8 points, and now try to choose them in general position (with respect to the property of containing the 8 points) so that the 9th point of intersection is in general position with regard to the $P_i$. This might seem plausibly possible if you don't think it through, but of course is in fact impossible, because the 8 given points uniquely determine the 9th one. (The possible $E_i$ lie in a pencil.) My impression is that the Italians made errors of that sort, but in much more subtle contexts.

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A technical objection: while B-N theory does fit the bill of far-sightedness but imprecision in algebraic geometry, both Alexander von Brill and Max Noether were in fact German... (And perhaps they were followers of Riemann in the way they thought about families of curves?) – Tim Perutz Mar 26 2010 at 13:51
"eight points"-->"eight points in P^2" of course. I see your point! I can believe that "generic" would have been a stumbling-block before we understood what a generic point really was. – Kevin Buzzard Mar 26 2010 at 13:51
PS @Tim: fair point! But of course my question wasn't specifically anti-italian ;-), I just wanted to see how one could make mistakes in algebraic geometry if one wasn't really too fussed about technical results in commutative algebra. – Kevin Buzzard Mar 26 2010 at 13:53
Dear Tim,  Thank you for pointing this out.  My understanding of this field, both now and historically, is pretty hazy; I was trying to remember a comment of Joe Harris (either made in person or in a paper; I don't remember which anymore) about an error of the Italians in studying the moduli of curves.  Now that I think again, I wonder if I am confusing a memory of reading Joe Harris with a memory of reading Mumford.  I will leave my response, since it may still be helpful for someone, but will add a disclaimer. – Emerton Mar 26 2010 at 18:03

I had the impression that there were false claims concerning rationality of certain Fano varieties, but I don't have any specific references on hand. For a more definite example, take a look the introduction to Mumford's "Rational equivalence of 0-cycles on surfaces". In this paper, he disproves something that Severi took as self evident.

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Ironically, Mumford used Severi's ideas to do so. – jvp Mar 26 2010 at 14:44

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Yes, which he acknowledges. To quote, "Now after criticizing Severi like this, I have to admit...". I've always admired Mumford's scholarship and sense of fairness. Thanks to which we have "Castelnuovo-Mumford" regularity... – Donu Arapura Mar 26 2010 at 15:13

Fano's list of 3-dimensional "Fano varieties" (so named by V.A.Iskovskikh) missed an entire class, of genus 12 if I recall correctly. This list was made complete later by Iskovskikh and Mukai-Umemura.

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Of course, we all know great mathematicians who constantly make mistakes even now, and not because of foundations.

In any case, it's not like "long dead Italian algebraic geometers" is a category of people who were all uniformly bad. For example, Enriques was notoriously careless, while Castelnuovo was much more scrupulous (I may be wrong, but as far as know he has not made any real mistake). I remember reading of a competion for a paper on resolution of singularities of surface; Castelnuovo and Enriques were in the committee. Beppo Levi presented his famous paper on the resolution of singularities for surfaces; Enriques asked him for a couple of examples and was convinced; Castelnuovo was not. The discussion got heated. Enriques exclaimed "I am ready to cut my head if this does not work" and Castelnuovo replied "I don't think that would prove it either".

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I should perhaps stress again that my question was most definitely not supposed to be an "anti-Italian" rant! It was an attempt to get some insight, on my part, of how one can do algebraic geometry badly if one doesn't have a big dollop of commutative algebra to back it up. Although I accepted damiano's answer, in truth I think Emerton told me the most: the point is that the closer a "generic point" is to a vague idea than to a non-closed point on a scheme, the more likely you are to be in trouble. – Kevin Buzzard Mar 28 2010 at 8:09

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I understand, and most certainly I was not offended. My point is that the general opinion that the old Italian algebraic geometers made mistakes because they did not have the proper foundations may be roughly right, but also simplistic. It it true that the Italian school went slowly astray, as discussed in Mumford's very interesting email message; but how much of it was due to personalities of the leaders of the school (particularly Severi) and how much to lack of proper foundations, I'll leave to others more competent than me to answer. – Angelo Mar 28 2010 at 10:31

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As I feel it (with not enough historical competence to prove it) in this and similar ages, the successive rigorous foundation came as an answer, after a call raised by the possibilities offered by new ideas and new fields that were disclosed to mathematicians. The same happened in different times with analysis and set theory. Mathematicians are not intrinsically rigorous; the rigor always came as a safety tool. – Pietro Majer Jun 24 2010 at 8:56

A beautiful survey article on the Italian school, with a discussion of several errors of all kinds by Severi, can be found in

• The Legacy of Niels Henrik Abel (Oslo 2002), Springer-Verlag 2004: Brigaglia, Ciliberto, Pedrini, The Italian school of algebraic geometry and Abel's legacy, 295--347
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There is also a longer article by Brigaglia and Ciliberto, "Italian algebraic geometry between the two world wars" (originally a chapter in a book on Italian mathematics of the interwar period), translated into English and published as Queen's Papers in Pure and Applied Mathematics, vol 100, 1995, Kingston, Ontario – Victor Protsak Aug 16 2010 at 4:51
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