Smooth threefolds in ${\Bbb P}^5$ without apparent triple or quadruple points and a quadruple-point formula

Abstract.

For a projective variety X of codimension 2 in ${\Bbb P}^{n+2}$ defined over the complex number field $\Bbb C$, 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 ${\Bbb P}^{n+2}$, equivalently, the locus of $(k+1)$-ple points of a generic projection of X to ${\Bbb P}^{n+1}$ is empty. We show that a smooth threefold in ${\Bbb P}^5$ 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 ${\Bbb P}^5$ and give necessary cohomological conditions for smooth threefolds in ${\Bbb P}^5$ 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 ${\Bbb P}^4$ 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.