what mistakes did the Italian algebraic geometers actually make? 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 mathematical 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.
 A: 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 65 nodes such as the Barth sextic (and this is optimal by Jaffe--Ruberman).
(*) 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, MR0025179, doi:10.1007/bf02418077.
A: 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.
A: 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

A: [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.
A: 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.
A: 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 competition 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".
