Geometry of dots and ropes
Author:
Karen A. Chandler
Journal:
Trans. Amer. Math. Soc. 347 (1995), 767784
MSC:
Primary 14H45; Secondary 14N05
MathSciNet review:
1273473
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Abstract: An dot is the first infinitesimal neighbourhood of a point with respect to an 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 higherdimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let be a collection of dots in uniform position in . Then the Hilbert function of satisfies for . Equality occurs for some with if and only if is contained in a rational normal curve , and the tangent directions to this curve at these points are all contained in . Equality occurs for some with if and only if is contained in the first infinitesimal neighbourhood of with respect to a subbundle, of rank and of maximal degree, of the normal bundle of in . This implies an upper bound on the degree of a subbundle of rank of the normal bundle of an irreducible nondegenerate smooth curve of degree in , by a Castelnuovo argument.
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 , Hilbert functions of dots in linear general position, Proc. Conference on ZeroDimensional Schemes, Ravello, Italy, June, 1992.
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 J. Harris (with D. Eisenbud), Curves in projective space, Les Presses de l'Université de Montréal, Montréal, Québec, Canada, 1982. MR 685427 (84g:14024)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512734735
PII:
S 00029947(1995)12734735
Article copyright:
© Copyright 1995
American Mathematical Society
