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It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the top of page 6, it is said that the problem is that showing a variety is not unirational is difficult, and in fact "there are no ways in practice" to do so.

Since these notes were published in 2001, and since a quick search didn't turn up anything, my question is: has there been any recent progress in this area? That is, are there any new methods for proving a given variety is not unirational? Or are there any recent cases in which a class of varieties is proven to be not unirational?

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    $\begingroup$ I think that if you find out you can put out a very good paper! $\endgroup$
    – IMeasy
    Commented May 18, 2013 at 20:29
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    $\begingroup$ This problem is still open. $\endgroup$
    – Jason Starr
    Commented May 19, 2013 at 13:42

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Just so there is an answer: the problem of equivalence / non-equivalence of rational connectedness and unirationality is still open.

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