On the irreducibility of Hilbert scheme of surfaces of minimal degree

Fedor Bogomolov; Viktor Kulikov

Open Mathematics (2013)

  • Volume: 11, Issue: 2, page 254-263
  • ISSN: 2391-5455

Abstract

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The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

How to cite

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Fedor Bogomolov, and Viktor Kulikov. "On the irreducibility of Hilbert scheme of surfaces of minimal degree." Open Mathematics 11.2 (2013): 254-263. <http://eudml.org/doc/268943>.

@article{FedorBogomolov2013,
abstract = {The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.},
author = {Fedor Bogomolov, Viktor Kulikov},
journal = {Open Mathematics},
keywords = {Hilbert scheme; Irreducible projective algebraic surfaces of minimal degree; irreducible surfaces of minimal degree; coverings of the plane},
language = {eng},
number = {2},
pages = {254-263},
title = {On the irreducibility of Hilbert scheme of surfaces of minimal degree},
url = {http://eudml.org/doc/268943},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Fedor Bogomolov
AU - Viktor Kulikov
TI - On the irreducibility of Hilbert scheme of surfaces of minimal degree
JO - Open Mathematics
PY - 2013
VL - 11
IS - 2
SP - 254
EP - 263
AB - The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.
LA - eng
KW - Hilbert scheme; Irreducible projective algebraic surfaces of minimal degree; irreducible surfaces of minimal degree; coverings of the plane
UR - http://eudml.org/doc/268943
ER -

References

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  1. [1] Ciliberto C., Flamini F., On the branch curve of a general projection of a surface to a plane, Trans. Amer. Math. Soc., 2011, 363(7), 3457–3471 http://dx.doi.org/10.1090/S0002-9947-2011-05401-2 Zbl1227.14022
  2. [2] Dolgachev I.V., Iskovskikh V.A., Finite subgroups of the plane Cremona group, In: Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, I, Progr. Math., 269, Birkhäuser, Boston, 2009, 443–548 Zbl1219.14015
  3. [3] Griffiths P., Harris J., Principles of Algebraic Geometry, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1978 Zbl0408.14001
  4. [4] Eisenbud D., Harris J., On varieties of minimal degree (a centennial account), In: Algebraic Geometry I, Brunswick, July 8–26, 1985, Proc. Sympos. Pure Math., 46(1), American Mathematical Society, Providence, 1987, 3–13 
  5. [5] Kulikov Vik.S., On Chisini’s conjecture, Izv. Math., 63(6), 1999, 1139–1170 (in Russian) http://dx.doi.org/10.1070/IM1999v063n06ABEH000267 Zbl0962.14005
  6. [6] Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian) http://dx.doi.org/10.1070/IM2008v072n05ABEH002423 Zbl1153.14012
  7. [7] Kulikov Vik.S., A remark on classical Pluecker’s formulae, preprint available at http://arxiv.org/abs/1101.5042 
  8. [8] Kulikov V.S., Kulikov Vik.S., On complete degenerations of surfaces with ordinary singularities in ℙ3, Sb. Math., 2010, 201(1), 129–158 http://dx.doi.org/10.1070/SM2010v201n01ABEH004068 
  9. [9] Reid M., Chapters on algebraic surfaces, In: Complex Algebraic Geometry, Park City, 1993, IAS/Park City Math. Ser., 3, American Mathematical Society, Providence, 1997 
  10. [10] Semple J.G., Roth L., Introduction to Algebraic Geometry, Oxford, Clarendon Press, 1949 Zbl0041.27903
  11. [11] Shafarevich I.R., Averbukh B.G., Vainberg Yu.R., Zhizhchenko A.B., Manin Yu.I., Moishezon B.G., Tyurina G.N., Tyurin A.N., Algebraic Surfaces, Trudy Mat. Inst. Steklov., 75, Nauka, Moscow, 1965 (in Russian) 

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