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Geometry of projection-generic space curves

Published online by Cambridge University Press:  01 July 2009

C. T. C. WALL
C. T. C. WALL*
Affiliation:
Department of Mathematical Science, University of Liverpool, L69 7ZL. e-mail: ctcw@liv.ac.uk
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Abstract

In earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.

To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.

Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.

This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 115 - 142
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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