Secant variety and syzygies of Hilbert scheme of two points

In this paper, we prove that $\mathrm{S}\mathrm{e}\mathrm{c}(X^{[2]})$ features the identifiability under the Grothendieck-Plücker embedding $X^{[2]}\hookrightarrow {\text{PP}}^N$ when $X$ is embedded by a $4$-very ample line bundle. We also prove that the embedding $X^{[2]}\hookrightarrow {\text{PP}}^N$ satisfies Green’s condition $(N_p)$ when the embedding of $X$ is positive enough. Accordingly, the singular locus of $\mathrm{S}\mathrm{e}\mathrm{c}(X^{[2]})$ is exactly $X^{[2]}$ 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 $\mathrm{S}\mathrm{e}\mathrm{c}(X^{[2]})$ when $X$ is a surface.