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.