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 -*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 higher-dimensional 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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DOI:
https://doi.org/10.1090/S0002-9947-1995-1273473-5

Article copyright:
© Copyright 1995
American Mathematical Society