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Transactions of the American Mathematical Society
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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1273473-5
PII: S 0002-9947(1995)1273473-5
Article copyright: © Copyright 1995 American Mathematical Society