Surfaces in $ {\mathbb{P}}^{4}$ whose 4-secant lines do not sweep out a hypersurface

José Carlos Sierra

doi:10.1016/j.crma.2013.09.016

Algebraic Geometry

Surfaces in

P^{4}

whose 4-secant lines do not sweep out a hypersurface
[Surfaces de

P^{4}

dont les droites quadrisécantes ne couvrent pas une hypersurface]

Présenté par : Claire Voisin

José Carlos Sierra ¹

¹ Instituto de Ciencias Matemáticas (ICMAT), Consejo Superior de Investigaciones Científicas (CSIC), Campus de Cantoblanco, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 623-625.

Résumé
Abstract

Nous montrons quʼune surface lisse dans $P^{4}$ dont les droites quadrisécantes ne couvrent pas une hypersurface de $P^{4}$ est, soit contenue dans un pinceau de cubiques, soit liée à une surface de Veronese via lʼintersection complète dʼune cubique et dʼune quartique.

We prove that a smooth surface in $P^{4}$ whose 4-secant lines do not sweep out a hypersurface of $P^{4}$ 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.

Reçu le : 2013-07-09
Accepté le : 2013-09-25
Publié le : 2013-08-01

DOI : 10.1016/j.crma.2013.09.016

Affiliations des auteurs :

José Carlos Sierra ¹

¹ Instituto de Ciencias Matemáticas (ICMAT), Consejo Superior de Investigaciones Científicas (CSIC), Campus de Cantoblanco, 28049 Madrid, Spain

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     author = {Jos\'e Carlos Sierra},
     title = {Surfaces in $ {\mathbb{P}}^{4}$ whose 4-secant lines do not sweep out a hypersurface},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {623--625},
     publisher = {Elsevier},
     volume = {351},
     number = {15-16},
     year = {2013},
     doi = {10.1016/j.crma.2013.09.016},
     language = {en},
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José Carlos Sierra. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.016/

Bibliographie
Cité par

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

[2] A.B. Aure The smooth surfaces in $P^{4}$ without apparent triple points, Duke Math. J., Volume 57 (1988), pp. 423-430

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

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

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

[6] P. De Poi Threefolds in $P^{5}$ with one apparent quadruple point, Commun. Algebra, Volume 31 (2003), pp. 1927-1947

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

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

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

[10] S. Kwak Smooth threefolds in $P^{5}$ without apparent triple or quadruple points and a quadruple-point formula, Math. Ann., Volume 320 (2001), pp. 649-664

[11] A. Lanteri On the existence of scrolls in $P^{4}$ , Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (8), Volume 69 (1980), pp. 223-227

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

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

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

[15] E. Mezzetti On quadrisecant lines of threefolds in $P^{5}$ , Le Matematiche, Volume 55 (2000), pp. 469-481 Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001)

[16] Ch. Okonek Flächen vom Grad 8 im $P^{4}$ , Math. Z., Volume 191 (1986), pp. 207-223

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

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

[19] F. Severi 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

Cité par Sources :

^☆ 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.

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