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.

**[AH]**J. Alexander and A. Hirschowitz,*La méthode d’Horace éclatée: application à l’interpolation en degré quatre*, Invent. Math.**107**(1992), no. 3, 585–602 (French). MR**1150603**, https://doi.org/10.1007/BF01231903**[ACGH]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932****[BE]**David Eisenbud and Mark Green,*Clifford indices of ribbons*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 757–765. MR**1273474**, https://doi.org/10.1090/S0002-9947-1995-1273474-7**[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]**David Eisenbud and Joe Harris,*Finite projective schemes in linearly general position*, J. Algebraic Geom.**1**(1992), no. 1, 15–30. MR**1129837****[EH2]**David Eisenbud and Joe Harris,*Schemes*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. The language of modern algebraic geometry. MR**1166800****[H1]**Joe Harris,*Curves in projective space*, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR**685427****[H2]**Joe Harris,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558**

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

Article copyright:
© Copyright 1995
American Mathematical Society