Abstract
The biregular geometry of punctual Hilbert schemes in dimensions 2 and 3, i.e., of schemes parametrizing fixed-length zero-dimensional subschemes supported at a given point on a smooth surface or a smooth three-dimensional variety, is studied. A precise biregular description of these schemes has only been known for the trivial cases of lengths 3 and 4 in dimension 2. The next case of length 5 in dimension 2 and the two first nontrivial cases of lengths 3 and 4 in dimension 3 are considered. A detailed description of the biregular properties of punctual Hilbert schemes and of their natural designularizations by varieties of complete punctual flags is given.
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References
J. Briançon, “Description de Hilbn C{x, y},”Invent. Math.,41, 45–89 (1977).
A. Iarrobino,Punctual Hilbert Schemes, Vol. 10, Memoires of the Amer. Math. Society (1977).
M. Granger, “Géométrie de schémas de Hilbert ponctuels,”Memoire Soc. Math. France Nouv. sér. n o9/10,111, No. 3, 1–84 (1981).
A. Iarrobino, “Compressed algebras and components of the punctual Hilbert scheme,” in:Sitges, 1983, Vol. 1124, Lecture Notes in Math, Springer, Berlin (1985), pp. 146–165.
A. Iarrobino, “Hilbert schemes of points: overview of last ten years,” in:Proc. of Amer. Math. Soc., Providence, R.I., Vol. 46, Symp. in Pure Math. (Algebraic Geometry, Bowdoin, 1985) (1987), pp. 297–320.
A. S. Tikhomirov, “A smooth model of punctual Hilbert schemes of a surface,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],208, 318–334 (1995).
M. Drezet and Potier J. Le, “Fibrés stables et fibrés exceptionnels surP 2,”Ann. scient. Éc. Norm. Sup., 4 série,18, 193–244 (1985).
C. BĂnicĂ, M. Putinar, and G. Schumacher, “Variation der globalen Ext and Deformationen kompakter komplexen Raume,”Math. Ann.,250, 135–155 (1980).
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Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 414–432, March, 2000.
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Tikhomirov, S.A. Punctual hilbert schemes of small length in dimensions 2 and 3. Math Notes 67, 348–364 (2000). https://doi.org/10.1007/BF02676671
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DOI: https://doi.org/10.1007/BF02676671