# 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.]

• 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. Mar 7, 2010 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.). Mar 7, 2010 at 3:39
• I would like to know enumerative algebraic geometry examples. Mar 7, 2010 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.) Mar 7, 2010 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.

• On 2005 Farkas Jensen and Payne showed that both $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$ are general type. On the same year Bruno and Verra showed that $\overline{\mathcal{M}}_{15}$ is rationally connected. Finally, on 2020 Farkas and Veraa showed that $\overline{\mathcal{M}}_{16}$ is not of general type. Feb 3, 2022 at 7:08

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.

• This doesn't seem to fulfill all of the conditions of the question, in that rigorous proofs now exist. Mar 7, 2010 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. Mar 7, 2010 at 7:11
• @Jose: 3264 non-singular ones (see example 9.1.9 in Fulton) Mar 7, 2010 at 13:03
• I guess we all hope that Fulton-McPherson are right... Mar 7, 2010 at 14:53
• @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. Mar 7, 2010 at 16:51

I think that the Italians geometers conjectured a generalization of Terracini's lemma which, if I am not mistaken, might be true but is still unknown.

Let $X \subset \mathbb{P}^N$ be a projetive variety over an algebraically closed field of characteristic $0$. Denote by $S(X) \subset \mathbb{P}^N$ the Zariski closure of the set of point lying on bisecant lines to $X$. The variety $S(X)$ is caled the secant variety to $X$. Let $x,y \in X$ be general points and let $z$ be a general point in $\langle x,y \rangle$ (the line joining $x$ and $y$), then we have:

$$T_{S(X),z} = \langle T_{X,x}, T_{X,y} \rangle,$$

where $T_{S(X),z}$ is the embedded tangent space to $S(X)$ at $z$ and $\langle T_{X,x}, T_{X,y} \rangle$ is the linear span in $\mathbb{P}^N$ of the tangent spaces to $X$ at $x$ and $y$.

This result, a consequence of the generic smoothness Theorem, is called Terracini's lemma and is due (probably with a non-rigorous proof) to Terracini (as the name suggests!).

There is a generalization of this result which was expected to be true be the Italian school. Denote by $\delta = 2 \dim X + 1 - \dim S(X)$. The number $\delta$ is called the secant-defect and is the difference between the expected and the actual dimensions of $S(X)$. If $x,y \in X$ are general points, Terracini's lemma immediately implies that: $$\dim (T_{X,x} \cap T_{X,y} ) = \delta - 1.$$

We denote by $S(X)_{stat}$ the Zariski closure of the set of points in $S(X)$ lying on secants $\langle x', y' \rangle$ such that $\dim (T_{X,x'} \cap T_{X,y'}) > \delta -1$. This variety is called the variety of stationnary bisecants to $X$.

I am pretty sure the Italian school considered the following fact to be true:

Fact : Let $x',y' \in X$ be general points such that $\langle x',y' \rangle \subset S(X)_{stat}$. Let $z$ be general in $\langle x',y' \rangle$, then:

$$T_{S(X)_{stat},z} = \langle T_{X,x'}, T_{X,y'} \rangle.$$

If $X \subset \mathbb{P}^N$ is a curve, then the above result is easily shown to be true. Indeed, if $X$ is not a line, then $S(X)_{stat}$ is necessarily a surface and since the generic tangent to $X$ is not a bitangent, we deduce that the result is true. I think it is also possible to prove this generalization of Terracini's lemma if $X$ is the closed orbit of a linear algebraic group acting linearly on $\mathbb{P}^N$. There are certainly many other cases where this generalization of Terracini's lemma is known to be true.

On the other hand, I am pretty sure this result is unknown in general. It might even drop into the box of the slightly false results, but which are still very interesting.