On quadrisecant lines of threefolds in P^5

TL;DR

This paper classifies smooth threefolds in P^5 with limited quadrisecant lines and explores conditions under which true quadrisecant lines exist without filling the space.

Contribution

It provides a complete classification of threefolds with only contained quadrisecant lines and characterizes those with true quadrisecant lines not filling the space.

Findings

Classified threefolds with only contained quadrisecant lines.

Proved that true quadrisecant lines imply the threefold is in a cubic hypersurface or contains certain plane curves.

Identified geometric conditions for the existence of non-filling quadrisecant lines.

Abstract

We study smooth threefolds of the projective space of dimension 5 whose quadrisecant lines don't fill up the space. We give a complete classification of those threefolds X whose only quadrisecant lines are the lines contained in X. Then we prove that, if X admits "true" quadrisecant lines, but they don't fill up the space, then either X is contained in a cubic hypersurface, or it contains a family of dimension at least two of plane curves of degree at least four.