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I am new to the study of unirational and rational varieties, but I want to know the motivation for why mathematicians started to study these conditions. The reasons that I could list to study unirational and rational varieties are the following.

  1. The study of unirational varieties can be used to answer questions about whether a scheme has a $ k $-rational point.
  2. In the minimal model program, rational varieties are in some sense the ones "most like $ \mathbb{P}^{n} $". Unirationality is a weakening of the rationality condition, but if we can answer questions about unirationality we may be able to a) strengthen the condition to deal with rationality and/or b) generalize the minimal model program by focusing on dominant, generically finite, rational maps from a variety $ X $ to a variety $ Y $ instead of looking at birational maps. Therefore, in order to answer questions about the minimal model program, or some kind of generalization, we would like to understand these conditions.

However, I wanted to pose the question to people who have studied these questions in detail, as the answers I gave to my own question seem too loose and I suspect that there is something that I am missing.

Thanks,

Schemer

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  • $\begingroup$ I am far from being an expert in these fields, but what I know is that it is really important that any reductive group G over any field is unirational. For example, it implies that $G(k)$ is Zariski-dense in $G$ for any non-finite field $k$. And it has some real applications in basic theory of reductive groups. If this is what you are looking for, then I can elaborate on this. $\endgroup$
    – gdb
    Commented Nov 30, 2017 at 7:36
  • $\begingroup$ J'anos Koll'ar has a lovely expository article, "Which are the simplest algebraic varieties?" He explains much of the history and motivation in that article. $\endgroup$
    – Jason Starr
    Commented Nov 30, 2017 at 10:32
  • $\begingroup$ @gdb You mean $k$-unirational (unirational is not enough). $\endgroup$
    – YCor
    Commented Dec 4, 2017 at 8:07
  • $\begingroup$ @YCor What do you mean by $k$-unirational and unirational? $\endgroup$
    – gdb
    Commented Dec 4, 2017 at 8:14
  • $\begingroup$ @gdb unirational means that there exists a dominant rational map from an affine space (defined a priori on some extension of a given definition field). A $k$-variety is $k$-unirational if there is a $k$-defined rational from an affine space. For instance, over the reals, the conic $x^2+y^2=-1$ is unirational but not $\mathbf{R}$-unirational (and indeed it has no real point!) $\endgroup$
    – YCor
    Commented Dec 4, 2017 at 12:08

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For a given variety it is important to know whether it is rational or unirational, because if this is the case, then the variety admits a nice parameterization. I believe this is the historical reason for studying these notions.

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