How to determine if an equation is a rational variety?

Question

Is there an easy way to determine if a given equation is a rational variety? For example, is $x^3+y^3+z^3=3$ a rational variety?

I have maxima installed on my computer, if that helps.

Answer 1

Since you tagged this question with number theory, it seems like you might be interested in the answer to this question over $\mathbf Q$ or other `arithmetic' fields. Let me just make a few comments about the problem over $\mathbf C$, to show that it is already extremely subtle there. In other words, the short answer to your question "is there an easy way?" is no.

Let's stick to equations that define smooth hypersurfaces in projective space. In your example, your equation can be rewritten in homogeneous form as $X^3+Y^3+Z^3-3W^3=0$, and it's easy to check that really is a smooth cubic surface in $\mathbf P^3$.

There are some general comments one can make. Let's say your defining equation has degree $d$ and involves $n$ variables. (I assume $n \geq 3$ to avoid degenerate cases.)

1. If $d=2$, your variety (a quadric hypersurface) is always rational. Proof: project from a point on the variety!

2. If $d \geq n$, then your hypersurface is never rational. One way to see this is that in this case, one can construct global differential forms of top degree on your variety, but a rational variety never has such forms.

3. If $3 \leq d \leq n-1$, the question is more difficult. Cubic surfaces in $\mathbf P^3$, including your example, are known (since the 19th century) always to be rational. By contrast, in the early 1970s it was proved that a smooth cubic threefold in $\mathbf P^4$ is never rational! And, as far as I know, it is still an open question whether every cubic hypersurface in $\mathbf P^n$ for $n \geq 5$ is rational. (Some of them are.)

Let me expand a little on the third point. Although rationality is a very difficult question in this case, there are some weaker properties that can be easier to deal with. For example, it is another classical fact that cubic hypersurfaces are always unirational, meaning that there is a finite dominant rational map $\mathbf P^n \dashrightarrow X$. (So $X$ can be "parameterised by rational functions", just not in a one-to-one fashion.) More generally, any hypersurface of degree $d \leq n-1$ is rationally connected, meaning that there is a rational curve connecting any two points. Depending on the context, sometimes these properties are adequate substitutes for rationality.

Finally, if indeed you are interested in the answers to these questions over non-closed fields such as $\mathbf Q$, then the whole story is even more complicated! For example, even a quadric surface need not be rational. I won't try to say anything in this case; instead I'll refer you to the book Rational and nearly rational varieties by János Kollár.