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Punctual hilbert schemes of small length in dimensions 2 and 3

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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

  1. J. Briançon, “Description de Hilbn C{x, y},”Invent. Math.,41, 45–89 (1977).

    Article  MATH  Google Scholar 

  2. A. Iarrobino,Punctual Hilbert Schemes, Vol. 10, Memoires of the Amer. Math. Society (1977).

  3. M. Granger, “Géométrie de schémas de Hilbert ponctuels,”Memoire Soc. Math. France Nouv. sér. n o9/10,111, No. 3, 1–84 (1981).

    Google Scholar 

  4. 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.

    Chapter  Google Scholar 

  5. 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.

    Google Scholar 

  6. 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).

    Google Scholar 

  7. M. Drezet and Potier J. Le, “Fibrés stables et fibrés exceptionnels surP 2,”Ann. scient. Éc. Norm. Sup., 4 série,18, 193–244 (1985).

    MATH  Google Scholar 

  8. C. BĂnicĂ, M. Putinar, and G. Schumacher, “Variation der globalen Ext and Deformationen kompakter komplexen Raume,”Math. Ann.,250, 135–155 (1980).

    Google Scholar 

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Authors and Affiliations

  1. K. D. Ushinskii Yaroslavl State Pedagogical University, USSR

    S. A. Tikhomirov

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  1. S. A. Tikhomirov
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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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