On the quadratic normality and the triple curve of three dimensional subvarieties of ${\mathbb P}^5$

TL;DR

This paper investigates the quadratic normality of smooth threefolds in projective 5-space and links non-normality to the reducibility of their triple curves, extending results to higher dimensions.

Contribution

It proves that non-quadratically normal threefolds in P^5 have reducible triple curves, providing new insights into the structure of such varieties and extending to higher dimensions.

Findings

Non-quadratically normal threefolds have reducible triple curves.

Results extend to higher-dimensional varieties.

Supports the conjecture on quadratic normality with new geometric conditions.

Abstract

A well-known conjecture asserts that smooth threefolds $X\subset\{\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If $X\subset\{\mathbb P}^5$ is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.