Congruences of lines in $\mathbb{P}^5$, quadratic normality, and completely exceptional Monge-Amp\`ere equations

TL;DR

This paper introduces two new families of sextic threefolds in projective 5-space, exploring their geometric properties and connections to Monge-Ampère equations, revealing novel examples in algebraic geometry.

Contribution

It constructs and analyzes two new families of sextic threefolds that are not quadratically normal, linking them to congruences of lines and Monge-Ampère equations.

Findings

Existence of two new families of sextic threefolds in P^5.

These threefolds are not quadratically normal.

One family arises from a smooth congruence of multidegree (1,3,3).

Abstract

The existence is proved of two new families of locally Cohen-Macaulay sextic threefolds in $\mathbb{P}^5$, 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-Amp\`ere 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.