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](http://mat.uab.es/~kock/RLN/rcv.pdf) 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? 



Proving a variety is not unirational
Just so there is an answer: the problem of equivalence / non-equivalence of rational connectedness and unirationality is still open.


