Journal of Topology

Journal of Topology Advance Access published online on February 5, 2008
Journal of Topology, doi:10.1112/jtopol/jtm015

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© 2008 London Mathematical Society

Projection genericity of space curves

C. T. C. Wall

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom ctcw{at}liv.ac.uk

If is a smooth space curve, we consider the family of projections of from a variable point not on to a fixed plane. For a residual set of curves , this family versally unfolds those singularities that occur in it.

To obtain a family of curves which is open as well as dense in the space of smooth maps, we must compactify the parameter space, so we study curves in real projective space, and include projections from points of the curve itself. If is smoothly embedded, the projection C_P of from P is a well-defined smooth curve, and for generic the family C_P has generic singularities.

However, when the point of projection moves off , the projection varies discontinuously. We define a family of plane curves, parametrised by the blow-up X of P³ along , such that for a point in the exceptional locus lying over P, we have the union of the projection C_P of from P and a straight line L through the image of the tangent at P. A key result asserts that this is a flat family. We give an explicit list of restrictions on the family C_P L (the key condition is that the total contact order of C_P with L never exceeds 2), and show that these hold for a dense open set of curves , and that if they do hold, there is a neighbourhood U of , such that the family of projections from points of U\ is generic.

Combining this list of conditions with those obtained previously gives a natural definition of a dense set of space curves , for which the complete family of projections has generic singularities, and we show that this set is also open.

Received September 6, 2007.

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2000 Mathematics Subject Classification 57R45 (primary), 51N15, 53A04 (secondary)

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Wall, C. T. C. Social Bookmarking          What's this?