Bogomolov, Fedor; Kulikov, Viktor S. On the irreducibility of Hilbert scheme of surfaces of minimal degree. (English) Zbl 1262.14004 Cent. Eur. J. Math. 11, No. 2, 254-263 (2013). The Hilbert scheme of irreducible surfaces of degree \(m\) in projective \({\mathbb P}^{m+1}\) is irreducible, except for \(m=4\) when the Hilbert scheme has two components. The article provides a new proof of this result using generic coverings of the projective plane. Reviewer: Roy Mikael Skjelnes (Stockholm) Cited in 1 Document MSC: 14C05 Parametrization (Chow and Hilbert schemes) Keywords:Hilbert scheme; irreducible surfaces of minimal degree; coverings of the plane PDFBibTeX XMLCite \textit{F. Bogomolov} and \textit{V. S. Kulikov}, Cent. Eur. J. Math. 11, No. 2, 254--263 (2013; Zbl 1262.14004) Full Text: DOI References: [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; · Zbl 1227.14022 [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; · Zbl 1219.14015 [3] Griffiths P., Harris J., Principles of Algebraic Geometry, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1978; · Zbl 0408.14001 [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] Kulikov Vik.S., On Chisini’s conjecture, Izv. Math., 63(6), 1999, 1139-1170 (in Russian) http://dx.doi.org/10.1070/IM1999v063n06ABEH000267; · Zbl 0962.14005 [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; · Zbl 1153.14012 [7] Kulikov Vik.S., A remark on classical Pluecker’s formulae, preprint available at http://arxiv.org/abs/1101.5042; · Zbl 1375.14100 [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; · Zbl 1205.14011 [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] Semple J.G., Roth L., Introduction to Algebraic Geometry, Oxford, Clarendon Press, 1949; · Zbl 0041.27903 [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); · Zbl 0154.21001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.