On the Variety of Complete Punctual Flags of Length 5 in Dimension 2

We consider the variety $X_d$ of complete punctual flags of length $d$ in dimension 2 defined as the closure of the variety of complete curvilinear zero-dimensional subschemes of length $\le d$ with support at the fixed point on a smooth algebraic surface; this closure is taken in the direct product of punctual Hilbert schemes. It is known that, for $2\le d\le 4$, the variety $X_d$ is smooth and coincides with the projectivization of the rank-2 vector bundle over $X_{d-1}$, where the bundle is described as the corresponding $\mathcal Ext$-sheaf. A similar bundle $\mathcal E$ is also defined over $X_4$. However, its projectivization $\mathbf P(\mathcal E)$ is birationally isomorphic but is not isomorphic to $X_5$. M. Gulbrandsen showed that $X_5$ has a curve of singularities. In the present article, we give a precise description of a minimal birational transformation of $X_5$ into $\mathbf P(\mathcal E)$ and interpret this transformation and the singularities of $X_5$ in terms of $\mathcal Ext$-sheaves.