It seems to me that the degree three equations have a special place among algebraic equations. Let me clarify what I have in mind.
- Degree 3 curves: The only curves that hold a group structure.
- 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.
- 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).
Here is my question. Is there a phenomena like this in some other degrees?
In other words, now I'm inclined to think that there is something very special about degree three equations, could this be true? or I'm just happen to be missing other equally important examples in other degrees?
Any answers/comments/examples will be appreciated.
Edit: sure, I deliberately didn't say anything about smoothness and very general surfaces. In order to keep the question simple, please put the sentence in context.