Secant variety and syzygies of Hilbert scheme of two points

In this paper, we prove that the singular locus of Sec (X[2]) coincides with X[2] under the Grothendieck-Pl\"ucker embedding X[2] PN when X is embedded by a $4$-very ample line bundle. We also prove that the embedding X[2] PN satisfies Green's condition (Nₚ) when the embedding of X is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to Sec(X[2]) when X is a surface.