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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Geometry of dots and ropes


Author: Karen A. Chandler
Journal: Trans. Amer. Math. Soc. 347 (1995), 767-784
MSC: Primary 14H45; Secondary 14N05
DOI: https://doi.org/10.1090/S0002-9947-1995-1273473-5
MathSciNet review: 1273473
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Abstract: An $ \alpha $-dot is the first infinitesimal neighbourhood of a point with respect to an $ (\alpha - 1)$-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that

Theorem 1. Let $ \Gamma $ be a collection of $ d$ $ \alpha $-dots in uniform position in $ {\mathbb{P}^n},\alpha \geqslant 2$. Then the Hilbert function $ {h_\Gamma }$ of $ \Gamma $ satisfies

$\displaystyle {h_\Gamma }(r) \geqslant \min (rn + 1,2d) + (\alpha - 2)\min ((r - 1)n - 1,\,d)$

for $ r \geqslant 3$. Equality occurs for some $ r$ with $ rn + 2 \leqslant 2d$ if and only if $ {\Gamma _{{\text{red}}}}$ is contained in a rational normal curve $ C$, and the tangent directions to this curve at these points are all contained in $ \Gamma $. Equality occurs for some $ r$ with $ (r - 1)n \leqslant d$ if and only if $ \Gamma $ is contained in the first infinitesimal neighbourhood of $ C$ with respect to a subbundle, of rank $ \alpha - 1$ and of maximal degree, of the normal bundle of $ C$ in $ {\mathbb{P}^n}$.

This implies an upper bound on the degree of a subbundle of rank $ \alpha - 1$ of the normal bundle of an irreducible nondegenerate smooth curve of degree $ d$ in $ {\mathbb{P}^n}$, by a Castelnuovo argument.


References [Enhancements On Off] (What's this?)

  • [AH] J. Alexander and A. Hirschowitz, La méthode d'Horace éclatée: application à l'interpolation en degré quatre, Invent. Math. 107 (1992), 585-602. MR 1150603 (93d:13017)
  • [ACGH] E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of algebraic curves, Vol. I, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
  • [BE] D. Bayer and D. Eisenbud, Ribbons and their canonical embeddings, Trans. Amer. Math. Soc. 347 (1995), 757-765. MR 1273474 (95g:14033)
  • [C1] K. Chandler, Hilbert functions of dots in uniform position, Ph.D. Thesis, Harvard Univ., 1992.
  • [C2] -, Hilbert functions of dots in linear general position, Proc. Conference on Zero-Dimensional Schemes, Ravello, Italy, June, 1992.
  • [EH1] D. Eisenbud and J. Harris, Finite projective schemes in linearly general position, J. Algebraic Geom. 1 (1992), 1-40. MR 1129837 (92i:14035)
  • [EH2] D. Eisenbud and J. Harris, Schemes: The language of algebraic geometry, Wadsworth and Brooks, Belmont, Calif., 1992. MR 1166800 (93k:14001)
  • [H1] J. Harris (with D. Eisenbud), Curves in projective space, Les Presses de l'Université de Montréal, Montréal, Québec, Canada, 1982. MR 685427 (84g:14024)
  • [H2] J. Harris, Algebraic geometry: A first course, Graduate Texts in Math., 33, Springer-Verlag, 1992. MR 1182558 (93j:14001)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1273473-5
Article copyright: © Copyright 1995 American Mathematical Society

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