On the quadratic normality and the triple curve of three dimensional subvarieties of P^5

A well-known conjecture asserts that smooth threefolds X in 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 in P^5 is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.