Linear syzygies and linear subspaces whose lines are multisecant

The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X\subseteq\mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\c