Irrational Complete Intersections

I will begin with a brief survey of the problem of rationality of complete intersections. Most of the talk will be devoted to an outline of the proof of the following statement, which builds upon work by Koll\'ar: The complete intersection of $r$ very general hypersurfaces of degrees $d_1,\dots, d_r$ in complex projective $N$-space is not ruled, and therefore not rational, provided that $\sum d_i \ge (3/4) N + 2r + 1$. The argument employs a coarse description of the singularities of generic maps in positive characteristics. It seems that further irrationality results could be possible if finer descriptions were available.