Generic space curves, geometry and numerology

Journal of Singularities
volume 2 (2010), 191-209
Proceedings of Singularities in Aarhus, August 2009

Received: 3 September 2009

DOI: 10.5427/jsing.2010.2l

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Abstract:

A projective curve Γ in P³(C) defines a stratification of P³ according to the types of the singularities of the projection of Γ from the variable point. In this paper we calculate the degrees of these strata, assuming that Γ is projection-generic in a particular sense. We use geometrical properties of the stratifications of P³ and of the blow-up B_Γ of P³ along Γ (with exceptional set denoted E_Γ) to introduce several auxiliary curves: more precisely, there are three 2-dimensional strata: the surface of tangents to Γ, the surface of T-secants (i.e. lines joining two points with coplanar tangents), and the surface of 3-secants. We obtain three plane curves by intersecting these with a generic plane. Three curves in E_Γ were introduced in some of our earlier work. We also have three curves in ΓxΓ closely related to them. Our numerical results are obtained by applying the genus and related formulae to these curves.

Keywords:

Mathematical Subject Classification:

14H50, 14N10

Author(s) information:

C. T. C. Wall