Algebraic Geometry
Surfaces in P4 whose 4-secant lines do not sweep out a hypersurface
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 623-625.

We prove that a smooth surface in P4 whose 4-secant lines do not sweep out a hypersurface of P4 either lies on a pencil of cubic hypersurfaces, or else is linked to a Veronese surface by the complete intersection of a cubic and a quartic hypersurface.

Nous montrons quʼune surface lisse dans P4 dont les droites quadrisécantes ne couvrent pas une hypersurface de P4 est, soit contenue dans un pinceau de cubiques, soit liée à une surface de Veronese via lʼintersection complète dʼune cubique et dʼune quartique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.09.016
Sierra, José Carlos 1

1 Instituto de Ciencias Matemáticas (ICMAT), Consejo Superior de Investigaciones Científicas (CSIC), Campus de Cantoblanco, 28049 Madrid, Spain
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Sierra, José Carlos. Surfaces in $ {\mathbb{P}}^{4}$ whose 4-secant lines do not sweep out a hypersurface. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 623-625. doi : 10.1016/j.crma.2013.09.016. http://www.numdam.org/articles/10.1016/j.crma.2013.09.016/

[1] Aure, A.B. On surfaces in projective 4-space, 1987 (Thesis, Oslo)

[2] Aure, A.B. The smooth surfaces in P4 without apparent triple points, Duke Math. J., Volume 57 (1988), pp. 423-430

[3] Bauer, I. Inner projections of algebraic surfaces: a finiteness result, J. Reine Angew. Math., Volume 460 (1995), pp. 1-13

[4] Bertolini, M.; Turrini, C. Surfaces in P4 with no quadrisecant lines, Beitr. Algebra Geom., Volume 39 (1998), pp. 31-36

[5] Cayley, A. On skew surfaces, otherwise scrolls, Philos. Trans. R. Soc. Lond., Volume 153 (1863), pp. 453-483

[6] De Poi, P. Threefolds in P5 with one apparent quadruple point, Commun. Algebra, Volume 31 (2003), pp. 1927-1947

[7] Gruson, L.; Peskine, Ch. Genre des courbes de lʼespace projectif, Univ. Tromsø, Tromsø, 1977 (Algebraic Geometry, Proc. Sympos.), Springer, Berlin (1978), pp. 31-59

[8] Ionescu, P. Embedded projective varieties of small invariants, III, LʼAquila, 1988 (Lect. Notes Math.), Volume vol. 1417, Springer, Berlin (1990), pp. 138-154

[9] Kleiman, S.L. Multiple-point formulas. I. Iteration, Acta Math., Volume 147 (1981), pp. 13-49

[10] Kwak, S. Smooth threefolds in P5 without apparent triple or quadruple points and a quadruple-point formula, Math. Ann., Volume 320 (2001), pp. 649-664

[11] Lanteri, A. On the existence of scrolls in P4, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (8), Volume 69 (1980), pp. 223-227

[12] Le Barz, P. Validité de certaines formules de géométrie énumérative, C. R. Acad. Sci. Paris, Sér. A, Volume 289 (1979), pp. 755-758

[13] Le Barz, P. Formules pour les multisécantes des surfaces, C. R. Acad. Sci. Paris, Sér. I, Volume 292 (1981), pp. 797-800

[14] Le Barz, P. Quelques formules multisécantes pour les surfaces, Sitges, 1987 (Enumerative Geometry), Springer, Berlin (1990), pp. 151-188

[15] Mezzetti, E. On quadrisecant lines of threefolds in P5, Le Matematiche, Volume 55 (2000), pp. 469-481 Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001)

[16] Okonek, Ch. Flächen vom Grad 8 im P4, Math. Z., Volume 191 (1986), pp. 207-223

[17] Ran, Z. On projective varieties of codimension 2, Invent. Math., Volume 73 (1983), pp. 333-336

[18] Ran, Z. The (dimension+2)-secant lemma, Invent. Math., Volume 106 (1991), pp. 65-71

[19] Severi, F. Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e aʼ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo, Volume 15 (1901), pp. 33-51

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Research supported by the “Ramón y Cajal” contract RYC-2009-04999 of MICINN, the project MTM2012-32670 and the ICMAT “Severo Ochoa” project SEV-2011-0087 of MINECO.