Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}\_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}\_Z$ the structure sheaf. Is it true that $$\mbox{Hom}\_X(\mathcal{I}\_Z,\mathcal{O}\_Z) \cong \mbox{Ext}^1\_X(\mathcal{I}\_Z, \mathcal{I}\_Z)?$$
Any hint or reference will be most welcome.



Tangent space to Hilbert schemes of points
See Lemma B.5.6 in A. Kuznetsov, Yu. Prokhorov, C. Shramov, "Hilbert schemes of lines and conics and automorphism groups of Fano threefolds", Japanese Journal of Mathematics, V. 13 (2018), N. 1, pp. 109-185.


