# Italian school of algebraic geometry and rigorous proofs

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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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
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
I would like to know enumerative algebraic geometry examples. – Petya Mar 7 '10 at 3:52
Petya, I'm with you, and you could ask this question in a week or two once this one has more-or-less wrapped up (to avoid having too many questions that are similar.) – Charles Siegel Mar 7 '10 at 17:10

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.