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Many of the amazing results by Italian geometers of the second half of the 19th and the first half of the 20th century were initially given heuristic explanations rather than rigorous proofs by their discoverers. Proofs appeared only later. In some cases, an intuitive explanation could be more or less directly translated into modern language. In some other cases, essentially new ideas were required (e.g., among others, the classification of algebraic surfaces by Shafarevich's seminar; construction of the moduli spaces of curves and their projective compactifications by Deligne, Mumford and Knudsen; solution of the Luroth problem by Iskovskikh and Manin).

I would like to ask: what are, in your opinion, the most interesting results obtained by pre-1950 Italian geometers which still do not have a rigorous proof?

[This is a community wiki, since there may be several answers, none of which is the "correct" one; however, please include as many things as possible per posting -- this is not intended as a popularity contest.]

[upd: since I'me getting much less answers that I had expected (in fact, only one so far), I would like to clarify a couple of things: as mentioned in the comments, I would be equally interested in results which are "slightly false" but are believed to be essentially correct, e.g. a classification with a particular case missing etc. I'm also interested in natural generalizations that still haven't been proven such as extending a result to finite characteristic etc.]

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Nice question!!! –  Petya Mar 7 '10 at 3:26
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Do you include "amazing results" that have turned out to be false, even "slightly"? Not to do so will dramatically reduce the number of responses, I think. –  Pete L. Clark Mar 7 '10 at 3:29
    
Petya -- thanks! –  algori Mar 7 '10 at 3:36
    
Pete -- yes, I would be interested in results that are "slightly false" but essentially correct (such as a classification with particular case or two missing etc.). –  algori Mar 7 '10 at 3:39
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I would like to know enumerative algebraic geometry examples. –  Petya Mar 7 '10 at 3:52
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Severi proved that the moduli space of curves $M_g$ is unirational when $g$ is at most $10$. This has now been made rigorous. Severi further conjectured that the moduli space is unirational for all values of $g$, but this was famously disproved by Eisenbud, Harris, and Mumford. They prove that $\overline{M}_g$ is of general type when $g \geq 24$. Farkas has shown that it is of general type when $g = 22$. It is known that when $g \leq 14$ the moduli space is unirational, but I believe that for remaining values of $g$, this problem is still open.

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The funniest example I know of is the number of conics tangent to five conics. There are now a number of different proofs, all based on modern intersection theory. Over the years, and until Fulton and MacPherson formalized/discovered/invented intersection theory people gave several wrong answers, as well as several wrong proofs of the correct one.

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This doesn't seem to fulfill all of the conditions of the question, in that rigorous proofs now exist. –  Pete L. Clark Mar 7 '10 at 7:07
    
@Pete: it does not, but it "almost" does: the problem is from the mid 19th century, there were probably about 10 different wrong solutions, and the Fulton-McPherson computed it correctly around 1970. –  David Lehavi Mar 7 '10 at 7:11
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@Jose: 3264 non-singular ones (see example 9.1.9 in Fulton) –  David Lehavi Mar 7 '10 at 13:03
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I guess we all hope that Fulton-McPherson are right... –  Mariano Suárez-Alvarez Mar 7 '10 at 14:53
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@Andrea: the right answer was given several times - with wrong arguments. The computation in GH starts at p. 749 and ends four pages later. The reason it is so long is that the computation is done on the cohomology ring of the blow-up of the double lines locus, which is done precisely in order to get around the intersection theoretic problems. –  David Lehavi Mar 7 '10 at 16:51
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