Remarks on defective Fano manifolds

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).