Congruences of lines in P^5, quadratic normality, and completely exceptional Monge–Ampère equations

The existence is proved of two new families of sextic threefolds in P^5, which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Ampère equations. One of these families comes from a smooth congruence of multidegree (1, 3, 3) which is a smooth Fano fourfold of index two and genus 9.