Existence in the smooth threefold case

Determine whether there exists a nondegenerate smooth threefold X contained in P^5 whose cubic equations vanish, (I_X)_3=0, but whose 4-secant-line locus is not all of P^5, S_4(X)≠P^5.

Background

This is the first of two concrete unresolved cases of the general multisecant classification problem. It asks whether the phenomenon exhibited in the paper by a singular threefold in P5 can occur for a smooth threefold.

The paper constructs a nondegenerate singular threefold in P5 with (I_X)_3=0 and S_4(X)≠P5, showing that the answer becomes positive after weakening smoothness to regularity in codimension one. The smooth case itself is explicitly left unresolved.

References

Is there any nondegenerate smooth $n$-fold $X\subseteq\pr$ satisfying S_{d}(X)\neq\pr\text{ and }(I_X)_{d-1}=0 when \begin{enumerate} \item $n=3$, $r=5$, $d=4$, or

— Linear syzygies and linear subspaces whose lines are multisecant  (2610.12078 - Han et al., 8 Oct 2026) in Section 1, Open Question \ref{question:conjecture}(1)