Implicitization of rational surfaces using toric varieties

Details Implicitization of rational surfaces using toric varieties Implicitization of rational surfaces using toric varieties

2004-01-28

arxiv.org/abs/math/0401403v2

Mathematics

Algebraic Geometry

28 pages, 1 figure. Numerous major revisions. New proof of method of moving surfaces. Paper accepted and to appear in Journal

Scientific paper

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored.

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