MEGA 2007 Effective Methods in Algebraic Geometry Strobl, Austria, June 25th - 29th http://www.ricam.oeaw.ac.at/mega2007

Abstract

Title	Generators of the ideal of an algebraic space curve
Keywords
Abstract	Abhyankar has shown that the ideal of a smooth space curve can be generated by three polynomials. In this paper we generalize Abhyankar's results to show that they hold for a larger class of curves, and we improve his constructions since we remove any geometric hypotheses of transversality and give explicit efficient algorithms. We produce generators for the ideal of a space curve by studying the connection between the space curve and a suitable plane projection. The space curves which we can handle are allowed to have singular points, but the local rings of these singular points are required to project isomorphically to the plane projection, whose singular locus can also contain additional ordinary double points arising from the projection. These are the same conditions imposed by Berry who studied some properties of Gr\"obner bases for such curves. Here we show that, assuming the space curve is integral and birational over its planar projection, we can recover the ideal of the space curve from the polynomial defining the plane curve and any rational function representing the space coordinate. As an intermediate step we produce a Gr\"obner basis for the space curve ideal consisting of four generators without any use of Buchberger's algorithm. We then show how to reduce these four to three generators. Assuming one begins with a plane curve, we also show how to construct a rational function which will lift it into 3-space, separating any specific subset of its double points. In particular for plane curves with only double points as singularities, our construction gives three generators for its non-singular model in 3-space.

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