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Transactions of the American Mathematical Society

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



Obstructions to deforming a space curve

Author: Daniel J. Curtin
Journal: Trans. Amer. Math. Soc. 267 (1981), 83-94
MSC: Primary 14D15
MathSciNet review: 621974
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Abstract: Mumford described a curve, $ \gamma $, in $ {{\mathbf{P}}^3}$ that has obstructed infinitesimal deformations (in fact the Hilbert scheme of the curve is generically nonreduced). This paper studies $ \gamma '{\text{s}}$ Hilbert scheme by studying deformations of $ \gamma $ in $ {{\mathbf{P}}^3}$ over parameter spaces of the form $ \operatorname{Spec} (k[t]/({t^n})),\,n = 2,\,3,\, \ldots $. Given a deformation of $ \gamma $ over $ \operatorname{Spec} (k[t]/({t^n}))$ one attempts to extend it to a deformation of $ \gamma $ over $ \operatorname{Spec} (k[t]/({t^{n + 1}}))$. If it will not extend, this deformation is said to be obstructed at the nth order.

I show that on a generic version of Mumford's curve, an infinitesimal deformation (i.e., a deformation over $ \operatorname{Spec} (k[t]/({t^2}))$) is either obstructed at the second order, or at no order, in which case we say it is unobstructed.

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Keywords: Projective space curves, infinitesimal deformations, obstructed deformations, deformations of cones, Hilbert scheme
Article copyright: © Copyright 1981 American Mathematical Society

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