Francesco Russo

The known results about the stable irrationality  of very general smooth Fano complete
intersections $X^n\subset\mathbb{P}^{n+c}$ of  dimension $n\geq 3$ and fixed type $(d_1,\ldots, d_c)$ pose
the natural problem of understanding under which conditions  there (might) exist  rational examples of that type/dimension, e.g. 
index one; index two; quartic fourfolds and fivefolds; etc.  

After introducing  the problem and after briefly recalling the state of the art, we shall present some results and geometric motivations pointing towards the statement of a possible solution.