Skip to main content
SpringerLink
Log in

Two dimensional scrolls contained in quadric cones in ℙ5

  • Published:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Aims and scope Submit manuscript

Abstract

Smooth, complex, ruled surfaces embedded in ℙ5 as linearly normal scrolls, such that they are contained in a quadric cone, are considered. Rational scrolls and some elliptic scrolls are shown to be the only ones contained in cones of rank 5. Results on scrolls contained in cones of lower ranks are also obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Alzati,M. Bertolini, andG. M. Besana, Projective normality of varieties of small degree.Quaderno del Dipartimento di Matematica — Universitá degli Studi di Milano, 1994.

  2. —, Projective normality of varieties of small degree.Comm. Algebra,5 (12) (1997), 3761–3771.

    Article  MathSciNet  Google Scholar 

  3. E. Arbarello, M. Cornalba, P. Griffiths, andJ. Harris,Geometry of Algebraic Curves, Volume I. Number 267 in Grundlehren der Mathematischen Wissenschaften. Springer Verlag, Berlin—Heidelberg—New York—Tokyo, 1985.

    Google Scholar 

  4. E. Arrondo andI. Sols, Classification of smooth congruences of low degree.J. Reine Angew. Math. 393 (1989), 199–219.

    MATH  MathSciNet  Google Scholar 

  5. G. M. Besana andS.DiRocco, On the projective normality of smooth surfaces of degree nine.Geom. Ded. 74 (1999), 1–21.

    Article  MATH  Google Scholar 

  6. G. Fano, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del quart’ordine.Comm. Math. Helv. 15 (1942), 71–80.

    Article  MathSciNet  Google Scholar 

  7. W. Fulton,Intersection Theory. Number 2 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Verlag, Berlin—Heidelberg—New York—Tokyo, 1984.

    Google Scholar 

  8. P. Griffiths andJ. Harris,Principles of Algebraic Geometry. John Wiley & Sons, 1978.

  9. M. Gross, The distribution of bidegrees of smooth surfaces in gr(1, ℙ3).Math. Ann. 292(1992), 127–147.

    Article  MATH  MathSciNet  Google Scholar 

  10. —, Surfaces of bidegree (3,n) ingr(l, ℙ3.).Mat. Z. 212 (1) (1993), 73–106.

    Article  MATH  Google Scholar 

  11. R. Hartshorne,Algebraic Geometry. Number 52 in GTM. Springer Verlag, New York—Heidelberg—Berlin, 1977.

    Google Scholar 

  12. P. Ionescu, Embedded projective varieties of small invariants III. In: Algebraic Geometry L’Aquila 1988. Number 1417 in Lecture Notes in Mathematics. Springer Verlag, 1990, pp. 138–154.

  13. P. Ionescu and M. Toma, On very ample vector bundles over curves.International Journal of Mathematics 8 (5) (1997), 633–643.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Ottaviani, On 3-folds in ℙ5 which are scrolls.Annali della Scuola Normale di Pisa Scienze Fisiche e Matematiche IV-XIX(3) (1992), 451–171.

    MathSciNet  Google Scholar 

  15. H. Tango, On (n-l)-dimensional projective spaces contained in the grassmann varietyGr(n, 1).J. Math. Kyoto Univ. 14 (3) (1974), 415–160.

    MATH  MathSciNet  Google Scholar 

  16. A. Verra, Smooth surfaces of degree nine in G(l, 3).Manuscripta Math. 62 (1988), 417–135.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Universita’ degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy

    A. Alzati

  2. Dipartimento di Matematica, Universita’ degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy

    M. Bertolini

  3. School of Computer Science, Telecommunications and Information Systems, DePaul University, 243 S. Wabash, 60604, Chicago IL, USA

    G. Besana

Authors
  1. A. Alzati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Bertolini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Besana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. Alzati, M. Bertolini or G. Besana.

Additional information

O. Riemenschneider

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alzati, A., Bertolini, M. & Besana, G. Two dimensional scrolls contained in quadric cones in ℙ5 . Abh.Math.Semin.Univ.Hambg. 74, 57–75 (2004). https://doi.org/10.1007/BF02941525

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02941525

2000 Mathematics Subject Classification

Key words and phrases

Search

Navigation