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 P n+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 P n+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.

Threefolds in ℙ5with One Apparent Quadruple Point

“…3 we prove two multiple point formulae: the quadruple point formula for a smooth threefold in P 5 and the formula which gives the number of 4-secant lines to a smooth surface in P 4 passing through a general point of the surface itself. I must say that after the proof of this formula, I realized that the same was obtained by Kwak (2001), but with other methods, i.e., through the monoidal construction. With this formula, Mezzetti has been able to exclude the only degree 12 smooth threefold in P 5 for which the existence was uncertain (see Edelmann, 1994).…”

SMOOTH n-DIMENSIONAL SUBVARIETIES OF ℙ2n-1 CONTAINING A FAMILY OF VERY DEGENERATE DIVISORS

“…Extending results of Severi [31] and Aure [3] on trisecant lines to surfaces in P 4 , Kwak [18] proved that a smooth threefold X ⊂ P 5 has no apparent triple points if and only if H 0 (I X (2)) = 0. A classification of threefolds with no apparent quadruple points is still unknown.…”

“…quadruple) points if S 3 (X) = P 5 (resp S 4 (X) = P 5 ). Threefolds in P 5 with no apparent triple points were classified by Kwak [18], extending to n = 3 the following result of Aure [3].…”

“…If dim Σ 3 (X) ≤ 2n − 2, then C ⊂ P 3 is a smooth curve with at most a finite number of trisecant lines and X ⊂ P n+2 is either a complete intersection of two quadric hypersurfaces or X ⊂ P 5 is the Segre embedding A classification of threefolds X ⊂ P 5 with no apparent quadruple points is still an open problem. If H 0 (I X (3)) = 0, then X ⊂ P 5 has no apparent quadruple points but it is not known if there exists such a threefold satisfying H 0 (I X (3)) = 0 (see [18] and [23]). Theorem 3.14 is useful to sharpen [23], Th.…”

On the Hilbert scheme of Palatini threefolds

“…The number q 4 ðX Þ of the 4-secant lines of X passing through a general point of P 5 is finite by [15]. Hence q 4 ðX Þ can be computed using the formula given by Kwak in [13] and it turns out that q 4 ðX Þ ¼ 1. So C 4 is the unique irreducible component of S 4 ðX Þ whose lines fill up P 5 .…”