Ellingsrud, Geir; Peskine, Christian Sur les surfaces lisses de \({\mathbb{P}}_ 4\). (On the smooth surfaces of \({\mathbb{P}}_ 4)\). (French) Zbl 0676.14009 Invent. Math. 95, No. 1, 1-11 (1989). Let \({\mathbb{P}}^ 4\) be the four-dimensional projective space over an algebraically closed field of characteristic zero. The authors prove that the smooth algebraic surfaces S in \({\mathbb{P}}^ 4\) satisfying the inequality \(K^ 2_ S\geq a\chi ({\mathcal O}_ S)\) for \(a\in {\mathbb{R}}\) and \(a<6\), are distributed in finitely many components of the Hilbert scheme of the smooth algebraic surfaces of \({\mathbb{P}}^ 4.\) It results as a corollary that the smooth rational surfaces of \({\mathbb{P}}^ 4\) describe finitely many components of the Hilbert scheme, as conjectured by Hartshorne and Lichtenbaum. Reviewer: E.Casas-Alvero Cited in 21 ReviewsCited in 32 Documents MSC: 14J10 Families, moduli, classification: algebraic theory 14C05 Parametrization (Chow and Hilbert schemes) 14N05 Projective techniques in algebraic geometry Keywords:projective four-space; components of the Hilbert scheme; smooth rational surfaces PDFBibTeX XMLCite \textit{G. Ellingsrud} and \textit{C. Peskine}, Invent. Math. 95, No. 1, 1--11 (1989; Zbl 0676.14009) Full Text: DOI EuDML References: [1] [A] Alexander, J.: Surfaces rationnelles non-sp?ciales dans ?4. Pr?publications, Univ. Nice, no 105, 1986 [2] [G.P] Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif. Algebraic Geometry. (687, pp. 31-59). Berlin-Heidelberg-New York: Springer 1977 [3] [H] Hartshorne, R.: Algebraic geometry. (Graduate Texts in Math., Vol. 52). Berlin-Heidelberg-New York: Springer 1977 [4] [M.1] Mumford, D.: Pathologies III. Am. J. Math.89, 94-104 (1967) · Zbl 0146.42403 [5] [M.2] Mumford, D.: Lectures on curves on an algebraic surface. Ann. Math. Stud.59, (1966) · Zbl 0187.42701 [6] [O] Okonek, C.: Fl?chen vom Grad 8 im ?4. Math. Z.191, 207-223 (1986) · Zbl 0611.14032 [7] [R] Roth, L.: On the projective classification of surfaces. Proc. London Math. Soc.42, 142-170 (1937) · Zbl 0015.26904 [8] [S] Severi, F.: Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a suoi punti tripli apparenti. Rend. Circ. Mat. Palermo, II. Ser.15, 33-51 (1901) · JFM 32.0648.04 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.