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Characterisations of lexicographic sets and simply-connected Hilbert schemes

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1255))

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Abstract

In the first part of the paper, we present two criteria to characterise lexicographic sets among Borel sets: one criterion by two combinatorial invariants of a Borel set, the other by an extremal property of a packing problem. In the second part, we apply these results to prove the simple-connectedness of certain Hilbert schemes by Gröbner basis theory.

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References

  1. M.F. Atiyah and I.G. Macdonald. Introduction to Commutative Algebra. Addison-Wesley Publishing Company, 1969.

    Google Scholar 

  2. D. A. Bayer. The division algorithm and the Hilbert scheme. PhD thesis, Harvard, 1982.

    Google Scholar 

  3. R. Braun and G. Floystad. A bound for the degree of smooth surfaces in P4 not of general type. Compositio Math., 93:211–229, 1994.

    Google Scholar 

  4. W. Bruns and J. Herzog. Cohen-Macaulay rings, volume 39 of Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  5. A.M. Bigatti. Upper bounds for the Betti numbers of a given Hilbert function. Communications in Algebra, 21:2317–2334, 1993.

    Google Scholar 

  6. M. Cook. An improved bound for the degree of smooth surfaces in P4 not of general type. Compositio Math., 102:141–145, 1996.

    Google Scholar 

  7. D. Eisenbud and J. Harris. Schemes: the language of modern algebraic geometry. Mathematics Series. Wadsworth and Brooks/Cole, Pacific Grove, 1992.

    Google Scholar 

  8. D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics. Springer, New York, 1995.

    Google Scholar 

  9. A. Galligo. Théoreme de division et stabilité en géométrie analytique locale. Ann. Inst. Fourier, 29:107–184, 1979.

    Google Scholar 

  10. G. Gotzmann. Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z., 158:61–70, 1978.

    Google Scholar 

  11. G. Gotzmann. Einige einfach-zusammenhängende Hilbertschemata. Math. Z., 180:291–305, 1982.

    Google Scholar 

  12. G. Gotzmann. A stratification of the Hilbert scheme of points on the projective plane. Math. Z., 199:539–547, 1988.

    Google Scholar 

  13. M. Green. Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann. In Proceedings of the Trento conference 1988, pages 147–154, 1988.

    Google Scholar 

  14. M. Green. Generic Initial Ideals. Lecture Notes, 1996.

    Google Scholar 

  15. G. Horrocks. Fixed point schemes of additive group actions. Topology, 8:233–242, 1969.

    Google Scholar 

  16. H. A. Hulett. Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Communications in Algebra, 21:2335–2350, 1993.

    Google Scholar 

  17. H. A. Hulett. A generalization of Macaulay's theorem. Communications in Algebra, 23:1249–1263, 1995.

    Google Scholar 

  18. J. Kollar. Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete; Folge 3, Band 32. Springer, Berlin, New York, 1996.

    Google Scholar 

  19. F. S. Macaulay. Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc., 26:531–555, 1927.

    Google Scholar 

  20. D. Mall. On the Castelnuovo Mumford regularity, submitted, 1996.

    Google Scholar 

  21. D. Mumford. Lectures on curves on an algebraic surface. Princeton University Press, Princeton, 1966.

    Google Scholar 

  22. K. Pardue. Nonstandard Borel-Fixed Ideals. PhD thesis, Brandeis, 1994.

    Google Scholar 

  23. A. A. Reeves. Combinatorial structure on the Hilbert scheme. PhD thesis, Cornell, 1992.

    Google Scholar 

  24. L. Robbiano. Term orderings on the polynomial ring. In B. F. Caviness, editor, Proc. EUROCAL 85, pages 513–517. Springer, 1985. Lect. Notes Comp. Sci., 204.

    Google Scholar 

  25. L. Robbiano. Introduction to the theory of Hilbert functions. In Anthony V. Geramita, editor, The Curves Seminar at Queen's, volume VII, pages B1–B26, Kingston, Ontario, 1990. Queen's University. Queen's Papers in Pure and Applied Mathematics, No. 85.

    Google Scholar 

  26. E. Sperner. Über einen kombinatorischen Satz von Macaulay und seine Anwendung auf die Theorie der Polynomideale, Abh.Math.Sem. Univ.Hamburg, 7:149–163, 1930.

    Google Scholar 

  27. V. Weispfenning. Admissible orders and linear forms. ACM SIGSAM Bulletin, 21:16–18, 1987.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, CH-8092, Zürich, Switzerland

    Daniel Mall

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  1. Daniel Mall
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Teo Mora Harold Mattson

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© 1997 Springer-Verlag Berlin Heidelberg

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Mall, D. (1997). Characterisations of lexicographic sets and simply-connected Hilbert schemes. In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_18

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  • DOI: https://doi.org/10.1007/3-540-63163-1_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63163-7

  • Online ISBN: 978-3-540-69193-8

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