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However, there is an interesting detail about cubic equations: It is expected that a very general cubic fourfold is unirational but not rational, but I believe this has not been proven yet.

In arbitrary dimension and degree we have some general results.

that a general hypersurface is unirational if its dimension is sufficiently large with respect to the degree. (There is an explicit formula for this). Since then there has been generalizations of this by Harris-Mazur-Pandharipande and Xi Chen.

It is also interesting to compare these to rational connectivity. Every Fano manifold is rationally connected, so Kollár's theorem implies that there are plenty of examples of rationally connected but not rational varieties.

The most interesting open problem in the area is that everyone expects that there are many rationally connected but not unirational varieties, but nobody knows an example.

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There are certainly some very nice properties of degree $3$ objects, but I agree with Chuck and Scott, that what you are observing is that this is the first non-trivial case of anything. On one hand that means that you will observe interesting behavior, on the other that there might be a larger literature dealing with this case.

Also, I can't help but comment on some of what you say in the question: You say:

## "Degree $3$ surfaces in $\mathbb P^3$: The only surfaces whose curves are not covered by the Noether-Lefchetz Theorem and therefore they are not complete intersection."

First of all the correct statement is the opposite: curves on cubic surfaces are not complete intersections and therefore the Noether-Lefschetz Theorem cannot be true. But this is an insignificant detail. Here $3$ is not that important. The main reason this happens is that all smooth cubics famously contain lines (27) and a curve of degree $c$ on a surface of degree $d>c$ cannot be a complete intersection. Also, the Noether-Lefschetz Theorem also fails for quadrics, that is degree $2$ surfaces: Their Picard group is rank $2$, so $\mathrm{Pic}(\mathbb P^3)\simeq \mathbb Z$ cannot surject onto it. So this isn't a real #$3$ mystery unless you count that $3$ is the largest degree where it fails, but then one could say that $4$ is the smallest where it holds, so $4$ is special.

Then again, you could add to your list: by the Lefschetz hyperplane theorem, the restriction of the Picard group to a hyperplane section is an isomorphism if the dimension is larger than $3$! So you could say dimension $3$ is the largest dimension where this fails. But this is really the same thing as the the Noether-Lefschetz thing.

And then you say:

## Threefold. Here, one can find examples (the only ones?) of unirrational varieties which are not rational. (can be done by looking at the intermediate jacobian).

There are at least two things wrong with this:

1. Those threefolds could be cubic (great!) but also quartic (that darn $4$ again).
2. This does not only happen in dimension three. János Kollár proved in this paper that

Theorem (Kollár): Let $X_d\subset \mathbb P^{n+1}$ be a very general hypersurface over $\mathbb C$ of degree $d\geq \frac 23(n + 3)$. Assume in addition that $n$ and $d$ are both even. Then $X_d$ is not (birationally) ruled.

Remark Obviously then $X_d$ is not rational.

On the other hand, Morin proved in

Morin, Ugo Sull'unirazionalità dell'ipersuperficie algebrica di qualunque ordine e dimensione sufficientemente alta. (Italian) Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, pp. 298–302 Edizioni Cremonense, Rome, 1942.

that a general hypersurface is unirational if its dimension is sufficiently large with respect to the degree. (There is an explicit formula for this). Since then there has been generalizations of this by Harris-Mazur-Pandharipande and Xi Chen.

Anyway, this gives plenty of examples of unirational but not rational varieties.

Finally, let me finish with the

## Group Law

This is indeed remarkably unique. As Scott pointed out we need to restrict to smooth cubic plane curves, but that is what you meant.

I'd like to remark that this is indeed about the number $3$ being special.

Consider curves of degree $d$ in $\mathbb P^2$. These are parametrized by a projective space $\mathbb P^N$ where $N= {d+2\choose 2}-1= \frac 12 d(d+3)$. Containing a fixed point of $\mathbb P^2$ imposes a linear condition on this space, so specifying $N$ points in general position determines a unique degree $d$ curve.

Generally two curves of degree $d$ will intersect in $d^2$ points. It turns out that $N=d^2$ for $d=3$ and this is why the group law defined by drawing lines is associative. Of course, there are more places where $3$ is needed, but those are sort of obvious reasons that one needs to even define the group law and this is a little bit more hidden.

Anyway, I should quit, but I want to say that my goal was not to take apart your question just to show that most of the nice properties you listed actually appear in more places.

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There are certainly some very nice properties of degree $3$ objects, but I agree with Chuck and Scott, that what you are observing is that this is the first non-trivial case of anything. On one hand that means that you will observe interesting behavior, on the other that there might be a larger literature dealing with this case.

Also, I can't help but comment on some of what you say in the question: You say:

## "Degree $3$ surfaces in $\mathbb P^3$: The only surfaces whose curves are not covered by the Noether-Lefchetz Theorem and therefore they are not complete intersection."

First of all the correct statement is the opposite: curves on cubic surfaces are not complete intersections and therefore the Noether-Lefschetz Theorem cannot be true. But this is an insignificant detail. Here $3$ is not that important. The main reason this happens is that all smooth cubics famously contain lines (27) and a curve of degree $c$ on a surface of degree $d>c$ cannot be a complete intersection. Also, the Noether-Lefschetz Theorem also fails for quadrics, that is degree $2$ surfaces: Their Picard group is rank $2$, so $\mathrm{Pic}(\mathbb P^3)\simeq \mathbb Z$ cannot surject onto it. So this isn't a real #$3$ mystery unless you count that $3$ is the largest degree where it fails, but then one could say that $4$ is the smallest where it holds, so $4$ is special.

Then again, you could add to your list: by the Lefschetz hyperplane theorem, the restriction of the Picard group to a hyperplane section is an isomorphism if the dimension is larger than $3$! So you could say dimension $3$ is the largest dimension where this fails. But this is really the same thing as the the Noether-Lefschetz thing.

And then you say:

## Threefold. Here, one can find examples (the only ones?) of unirrational varieties which are not rational. (can be done by looking at the intermediate jacobian).

There are at least two things wrong with this:

1. Those threefolds could be cubic (great!) but also quartic (that darn $4$ again).
2. This does not only happen in dimension three. János Kollár proved in this paper that

Theorem (Kollár): Let $X_d\subset \mathbb P^{n+1}$ be a very general hypersurface over $\mathbb C$ of degree $d\geq \frac 23(n + 3)$. Assume in addition that $n$ and $d$ are both even. Then $X_d$ is not (birationally) ruled.

Remark Obviously then $X_d$ is not rational.

On the other hand, Morin proved in

Morin, Ugo Sull'unirazionalità dell'ipersuperficie algebrica di qualunque ordine e dimensione sufficientemente alta. (Italian) Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, pp. 298–302 Edizioni Cremonense, Rome, 1942.

that a general hypersurface is unirational if its dimension is sufficiently large with respect to the degree. (There is an explicit formula for this). Since then there has been generalizations of this by Harris-Mazur-Pandharipande and Xi Chen.

Anyway, this gives plenty of examples of unirational but not rational varieties.

Finally, let me finish with the

## Group Law

This is indeed remarkably unique. As Scott pointed out we need to restrict to smooth cubic plane curves, but that is what you meant.

I'd like to remark that this is indeed about the number $3$ being special.

Consider curves of degree $d$ in $\mathbb P^2$. These are parametrized by a projective space $\mathbb P^N$ where $N= {d+2\choose 2}-1= \frac 12 d(d+3)$. Containing a fixed point of $\mathbb P^2$ imposes a linear condition on this space, so specifying $N$ points in general position determines a unique degree $d$ curve.

Generally two curves of degree $d$ will intersect in $d^2$ points. It turns out that $N=d^2$ for $d=3$ and this is why the group law defined by drawing lines is associative. Of course, there are more places where $3$ is needed, but those are sort of obvious reasons that one needs to even define the group law and this is a little bit more hidden.

Anyway, I should quit, but I want to say that my goal was not to take apart your question just to show that most of the nice properties you listed actually appear in more places.