×

zbMATH — the first resource for mathematics

On 3-folds in \(\mathbb{P}^ 5\) which are scrolls. (English) Zbl 0786.14026
The author studies the 3-folds \(X\) of degree \(d\) in \(\mathbb{P}^ 5\) which are scrolls over a surface \(E\). There are 4 examples classically known of such scrolls of degree \(d=3\), 6, 7, 9: the author proves that these are the only possible cases by analysing the genus of the general curve section. Moreover he studies the general embedding of a \(\mathbb{P}^ 1\)- bundle in \(\mathbb{P}^ 5\).

MSC:
14J30 \(3\)-folds
14N05 Projective techniques in algebraic geometry
14E25 Embeddings in algebraic geometry
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] A.B. Aure , On surfaces in projective 4-space , Thesis, Oslo 1987 .
[2] R. Braun - G. Ottaviani - M. Schneider - F.O. Schreyer , Boundedness for non-general type 3-folds in P5 , to appear in the Proceedings of CIRM Meeting ”Analysis and Geometry X”, Trento 1991 .
[3] R. Braun - G. Ottaviani - M. Schneider - F.O. Schreyer , Classification of log-special 3-folds in P5 , to appear.
[4] M. Beltrametti - A.J. Sommese , Comparing the classical and the adjunction theoretic definition of scrolls , to appear in the Proceedings of the 1990 Cetraro Conference ”Geometry of Complex Projective Varieties”. MR 1225588 | Zbl 0937.14027 · Zbl 0937.14027
[5] M. Beltrametti - A.J. Sommese , New properties of special varieties arising from adjunction theory , J. Math. Soc. Japan 43 , 381 - 412 ( 1991 ). Article | MR 1096439 | Zbl 0754.14027 · Zbl 0754.14027 · doi:10.2969/jmsj/04320381 · minidml.mathdoc.fr
[6] M. Beltrametti - M. Schneider - A.J. Sommese , Threefolds of degree 9 and 10 in P5 , Math. Ann. 288 , 613 - 644 ( 1990 ). MR 1079870 | Zbl 0723.14033 · Zbl 0723.14033 · doi:10.1007/BF01444540 · eudml:164742
[7] G. Castelnuovo , Ricerche di geometria della retta nello spazio a quattro dimensioni , Atti R. Ist. Veneto Sc. , ser. VII , 2 , 855 - 901 ( 1891 ). JFM 23.0865.01 · JFM 23.0865.01
[8] M.C. Chang , A filtered Bertini-type theorem , J. Reine Angew. Math. 397 , 214 - 219 ( 1989 ). MR 993224 | Zbl 0663.14008 · Zbl 0663.14008 · crelle:GDZPPN002206641 · eudml:153138
[9] M.C. Chang , Classification of Buchsbaum subvarieties of codimension 2 in projective space , J. Reine Angew. Math. 401 , 101 - 112 ( 1989 ). MR 1018055 | Zbl 0672.14026 · Zbl 0672.14026 · doi:10.1515/crll.1989.401.101 · crelle:GDZPPN002207001 · eudml:153173
[10] G. Ellingsrud - C. Peskine , Sur les surfaces lisses de P4 , Invent. Math. 95 , 1 - 11 ( 1989 ). MR 969410 | Zbl 0676.14009 · Zbl 0676.14009 · doi:10.1007/BF01394141 · eudml:143642
[11] W. Fulton , Intersection theory , Springer , Berlin 1984 . MR 732620 | Zbl 0541.14005 · Zbl 0541.14005
[12] M. Fiorentini - A.T. Lascu , Una formula di geometria numerativa , Ann. Univ. Ferrara , Sez. VII , 27 , 201 - 227 ( 1981 ). MR 653873 | Zbl 0513.14036 · Zbl 0513.14036
[13] A. Holme - H. Roberts , On the embeddings of Projective Varieties , Lecture Notes in Math. 1311 , 118 - 146 , Springer , Berlin 1988 . MR 951644 | Zbl 0663.14009 · Zbl 0663.14009
[14] S.L. Kleiman , Geometry on Grassmannians and applications to splitting bundles and smoothing cycles , Publ. Math. IHES 36 , 281 - 297 ( 1969 ). Numdam | MR 265371 | Zbl 0208.48501 · Zbl 0208.48501 · doi:10.1007/BF02684605 · numdam:PMIHES_1969__36__281_0 · eudml:103895
[15] A. Lanteri , On the existence of scrolls in P4 , Atti Accad. Naz. Lincei ( 8 ) 69 , 223 - 227 ( 1980 ). MR 670824 | Zbl 0509.14042 · Zbl 0509.14042
[16] A. Lanteri - C. Turrini , Some formulas concerning nonsingular algebraic varieties embedded in some ambient variety , Atti Accad. Sci. Torino 116 , 463 - 474 ( 1982 ). MR 840740 | Zbl 0606.14009 · Zbl 0606.14009
[17] N.Y. Netsvetaev , Projective varieties defined by small number of equations are complete intersections , in ”Topology and geometry”, Rohlin Sem. 1984-1986 , Lecture Notes in Math. 1346 , 433 - 453 , Springer , Berlin 1988 . MR 970088 | Zbl 0664.14028 · Zbl 0664.14028
[18] C. Okonek , Über 2-codimensionale Untermannigfaltigkeiten vom Grad 7 in P4 und P5 , Math. Z. 187 , 209 - 219 ( 1984 ). MR 753433 | Zbl 0575.14030 · Zbl 0575.14030 · doi:10.1007/BF01161705 · eudml:173484
[19] C. Okonek - M. Schneider - H. Spindler , Vector bundles on complex projective spaces , Birkhäuser , Boston 1980 . MR 561910 | Zbl 0438.32016 · Zbl 0438.32016
[20] F. Palatini , Sui sistemi lineari di complessi lineari di rette nello spazio a cinque dimensioni , Atti Ist. Veneto , 602 , 371 - 383 ( 1900 ). JFM 32.0553.04 · JFM 32.0553.04
[21] C. Peskine - L. Szpiro , Liaison des variétés algébriques I , Invent. Math. 26 , 271 - 302 ( 1974 ). MR 364271 | Zbl 0298.14022 · Zbl 0298.14022 · doi:10.1007/BF01425554 · eudml:142303
[22] M. Schneider , Vector bundles and low-codimensional submanifolds of projective space: a problem list, Topics in algebra . Banach Center Publications , vol. 26 , PWN Polish Scientific Publishers , Warsaw 1989 . MR 1171271 | Zbl 0739.14004 · Zbl 0739.14004
[23] A.J. Sommese , On the adjunction theoretic structure of projective varieties, in ”Complex Analysis and Algebraic Geometry” , Proceedings Göttingen 1985, Lecture Notes in Math. 1194 , 175 - 213 , Springer , Berlin 1986 . MR 855885 | Zbl 0601.14029 · Zbl 0601.14029 · doi:10.1007/BFb0077004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.