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



Why study unirational and rational varieties?
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. 


