A characterization of the complex projective plane (Q1902494)

The author gives the following interesting characterization of the projective plane, a ``first order analogue of Reiss condition''. Let \(S\) be a smooth projective surface over the complex field. Let \(D\) be a smooth irreducible curve on \(S\). Assume that either \(2D\) is very ample on \(S\) or \(S\) is minimal and \(D^2 >0\). Let \(D(1)\) be the first infinitesimal neighborhood of \(D\), \(D(1) = (D, {\mathcal O}_S/I^2)\), where \(I\) is the defining sheaf of \(D\) in \(S\). Then the restriction morphism \(\text{Pic} (S) \to \text{Pic} (D(1))\) is surjective if and only if \(S\) is the projective plane and \(D\) is a line. It would be an interesting question to extend the characterization above to higher dimensions.