On the quadratic normality and the triple curve of three dimensional   subvarieties of ${\mathbb P}^5$
  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. 
Pietro De Poi, Emilia Mezzetti, Jos\'e Carlos Sierra