  We prove that a complete intersection of $c$ very general hypersurfaces of
degree at least two in $N$-dimensional complex projective space is not ruled
(and therefore not rational) provided that the sum of the degrees of the
hypersurfaces is at least $\tfrac{2}{3} N + c + 1$. To this end we consider a
degeneration to positive characteristic, following Koll\'ar. Our argument does
not require a resolution of the singularities of the special fiber of the
degeneration. It relies on a generalization of Koll\'ar's "algebraic Morse
lemma" that controls the dimensions of the second-order Thom-Boardman
singularities of general sections of Frobenius pullbacks of vector bundles.
