Smooth threefolds in P-5 without apparent triple or quadruple points and a quadruple-point formula

Abstract

For a projective variety X of codimension 2 in Pn+2 defined over the complex number field C, it is traditionally said that X has no apparent (k + 1)-ple points if the (k + 1)-secant lines of X do not fill up the ambient projective space P-n+2, equivalently, the locus of (k + 1)-ple points of a generic projection of X to Pn+1 is empty. We show that a smooth threefold in P-5 has no apparent triple points if and only if it is contained in a quadric hypersurface. We also obtain an enumerative formula counting the quadrisecant lines of X passing through a general point of P-5 and give necessary cohomological conditions for smooth threefolds in P-5 without apparent quadruple points. This work is intended to generalize the work of F. Severi [fSe] and A. Aure [Au], where it was shown that a smooth surface in P-4 has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric. Furthermore we give open questions along these lines.