Projection genericity of space curves

Abstract

If Γ is a smooth space curve, we consider the family of projections of Γ from a variable point not on Γ to a fixed plane. For a residual set of curves Γ, 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 Γ is smoothly embedded, the projection CP of Γ from P ∈ Γ is a well-defined smooth curve, and for generic Γ the family CP has generic singularities. However, when the point of projection moves off Γ, the projection varies discontinuously. We define a family of plane curves, parametrised by the blow-up X of P3 along Γ, such that for a point in the exceptional locus lying over P ∈ Γ , we have the union of the projection CP of Γ 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 CP ∪ L (the key condition is that the total contact order of CP with L never exceeds 2), and show that these hold for a dense open set of curves Γ, and that if they do hold, there is a neighbourhood U of Γ, such that the family of projections from points of U \ Γ is generic. Combining this list of conditions with those obtained previously gives a natural definition of a dense set of space curves Γ, for which the complete family of projections has generic singularities, and we show that this set is also open.