Classification of smooth varieties with non-full multisecant locus

Classify nondegenerate smooth n-dimensional projective varieties X contained in projective r-space such that the degree-(d−1) component of the homogeneous ideal I_X vanishes, (I_X)_{d−1}=0, while the locus S_d(X) of d-secant lines does not equal the entire ambient projective space, S_d(X)≠P^r.

Background

The paper studies loci of d-secant lines to projective varieties and their relationship with syzygies, projections, and Castelnuovo–Mumford regularity. The condition (I_X)_{d−1}=0 excludes the trivial situation in which a nonzero equation of degree d−1 automatically forces S_d(X) to lie in a hypersurface and hence prevents it from filling the ambient space.

The authors formulate a general classification problem for smooth n-folds whose d-secant lines fail to fill the ambient projective space despite the absence of equations of degree d−1. They note that this problem is connected to the Eisenbud–Goto conjecture and that the smooth case of that conjecture remains open. The paper subsequently resolves two singular or normal analogues, but does not resolve the stated smooth classification problem.

References

Classify nondegenerate smooth $n$-folds $X\subseteq\pr$ such that $(I_X)_{d-1}=0$ and $S_d(X)\neq\pr$.

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

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 \item $n=4$, $r=7$, $d=3$?

— 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}(2)