Geodesic
On the Variety of Complete Punctual Flags of Length 5 in Dimension 2
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 277-282 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. S. Tikhomirov and S. A. Tikhomirov},
     title = {On the {Variety} of {Complete} {Punctual} {Flags} of {Length} 5 in {Dimension} 2},
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A. S. Tikhomirov; S. A. Tikhomirov. On the Variety of Complete Punctual Flags of Length 5 in Dimension 2. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 277-282. http://geodesic.mathdoc.fr/item/TM_2004_246_a18/

[1] Briançon J., “Description de $\mathrm{Hilb}^n\mathbf{C}\{x,y\}$”, Invent. Math., 41 (1977), 45–89 | DOI | MR | Zbl

[2] Granger M., Géométrie des schémas de Hilbert ponctuels, Mem. Soc. Math. France. Nouv. Sér., 8, Soc. Math. France, Paris, 1983 | MR | Zbl

[3] Iarrobino A., Punctual Hilbert schemes, Mem. AMS, 188, Amer. Math. Soc., Providence, RI, 1977 | MR

[4] Tikhomirov A. S., “Gladkaya model punktualnykh skhem Gilberta poverkhnosti”, Tr. MIAN, 208, 1995, 318–334 | MR | Zbl

[5] Tikhomirov S. A., “Punktualnye skhemy Gilberta maloi dliny v razmernostyakh 2 i 3”, Mat. zametki, 67:3 (2000), 414–432 | MR | Zbl

[6] Gulbrandsen M. G., A partial resolution of the punctual Hilbert scheme of a nonsingular surface, Preprint, Univ. Oslo, 2003 | MR