Skip to main content
SpringerLink
Log in

Datagraphs in Algebraic Geometry and K3 Surfaces

  • Conference paper
  • First Online:
Book cover Symbolic and Numerical Scientific Computation (SNSC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2630))

  • 525 Accesses

Abstract

Datagraphs are combinatorial graphs having database items at their vertices and geometric relationships along their edges. I describe their applicability to lists of examples in algebraic geometry generated by computer algebra, and illustrate this with a list of K3 surfaces as the database items. The main point is that when analysing a single surface during construction of the database, the datagraph makes available its close relations, and that very often these provide extra information required to complete the analysis.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Lists of K3 surfaces are available from http://www.maths.warwick.ac.uk/~gavinb/k3graph.html and k3db.html

  2. Altmok, S. (1998): Graded rings corresponding to polarised K3 surfaces and ℚ-Fano 3-folds. Thesis, Univ. of Warwick 93 + vii pp.

    Google Scholar 

  3. S. Altmok, Hilbert series and applications to graded rings, to appear in International journal of Mathematics and Mathematical science

    Google Scholar 

  4. S. Altmok, G. Brown and M. Reid, Fano 3-folds, K3 surfaces and graded rings, to appear in Contemp Math (SISTAG 2001 Proceedings, ed Berrick et al.)

    Google Scholar 

  5. A. Beauville, Complex algebraic surfaces. Second edition. LMS Student Texts, 34. CUP, 1996. x+132 pp

    Google Scholar 

  6. S-M. Belcastro, Picard lattices of families of K3 surfaces, Comm. Algebra 30 (2002), no. 1, 61–82

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Brown and M. Reid, Extreme K3 surfaces with high unprojections, in preparation

    Google Scholar 

  8. A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175–258

    Google Scholar 

  9. A. Durfee, Fifteen characterisations of Du Val singularities, Enseign. Math. (2) 25 (1979), no. 1–2, 131–163

    MathSciNet  Google Scholar 

  10. A.R. Iano-Fletcher, Working with weighted complete intersections, in Explicit birational geometry of 3-folds, CUP 2000, pp. 101–173

    Google Scholar 

  11. J. Johnson and J. Kollár, Fano hypersurfaces in weighted projective 4-spaces, Experiment. Math. 10 (2001), no. 1, 151–158

    MATH  MathSciNet  Google Scholar 

  12. M. Kreuze and H. Skarke, Lists of K3 and Calabi-Yau on the web at http://www.hep.itp.tuwien.ac.at/~kreuzer/CY/

  13. Magma (John Cannon’s computer algebra system): W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997) 235–265. See also http://www.maths.usyd.edu.au:8000/u/magma

    Article  MATH  Google Scholar 

  14. S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, J. algebraic geometry (to appear), preprint math.AG/0011094, 15 pp

    Google Scholar 

  15. M. Reid, Surfaces with p g = 0, K 2 = 1. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), No. 1, 75–92

    MATH  MathSciNet  Google Scholar 

  16. M. Reid, Chapters on algebraic surfaces. Complex algebraic geometry (Park City, UT, 1993), 3–159, IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, RI, 1997

    Google Scholar 

  17. M. Reid, Graded rings and birational geometry, in Proc. of algebraic geometry symposium (Kinosaki, Oct 2000), K. Ohno (Ed.), 1–72, available from http://www.maths.warwick.ac.uk/~miles/3folds

  18. I. Shimada and D-Q. Zhang, Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces, Nagoya Math. J. 161 (2001), 23–54

    MATH  MathSciNet  Google Scholar 

  19. B. Szendrői, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem. Internat. J. Math. 11 (2000), No. 3, 449–459.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Gavin Brown

Authors
  1. Gavin Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RISC-Linz, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Franz Winkler

  2. Institute of Computational Mathematics, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Ulrich Langer

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brown, G. (2003). Datagraphs in Algebraic Geometry and K3 Surfaces. In: Winkler, F., Langer, U. (eds) Symbolic and Numerical Scientific Computation. SNSC 2001. Lecture Notes in Computer Science, vol 2630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45084-X_9

Download citation

  • DOI: https://doi.org/10.1007/3-540-45084-X_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40554-2

  • Online ISBN: 978-3-540-45084-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation