119 research outputs found
Equations of parametric surfaces with base points via syzygies
Let S be a tensor product parametrized surface in P ; that is, S is given as the image of φ : P × P → P . This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This work extends the algorithm provided by Cox [Cox, D.A., 2001. Equations of parametric curves and surfaces via syzygies. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics vol. 286, pp. 1-20] for when φ has no base points, and it is analogous to some of the results of Busé et al. [Busé, L., Cox, D., D\u27Andrea, C., 2003. Implicitization of surfaces in P in the presence of base points. J. Algebra Appl. 2 (2), 189-214] for the case of a triangular parametrization φ : P → P with base points. © 2004 Elsevier Ltd. All rights reserved. 3 1 1 3 3 2
Equations of parametric surfaces with base points via syzygies
Suppose is a parametrized surface in complex projective 3-space given as the image of . The implicitization problem is to compute an implicit equation of using the parametrization . An algorithm using syzygies exists for computing if has no base points, i.e. is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4
Matrix representations for toric parametrizations
In this paper we show that a surface in P^3 parametrized over a 2-dimensional
toric variety T can be represented by a matrix of linear syzygies if the base
points are finite in number and form locally a complete intersection. This
constitutes a direct generalization of the corresponding result over P^2
established in [BJ03] and [BC05]. Exploiting the sparse structure of the
parametrization, we obtain significantly smaller matrices than in the
homogeneous case and the method becomes applicable to parametrizations for
which it previously failed. We also treat the important case T = P^1 x P^1 in
detail and give numerous examples.Comment: 20 page
Implicitization of rational surfaces using toric varieties
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.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of
moving surfaces. Paper accepted and to appear in Journal of Algebr
Implicitization of Bihomogeneous Parametrizations of Algebraic Surfaces via Linear Syzygies
We show that the implicit equation of a surface in 3-dimensional projective
space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a
given positive integer d, can be represented and computed from the linear
syzygies of its parametrization if the base points are isolated and form
locally a complete intersection
Compactifications of rational maps, and the implicit equations of their images
In this paper we give different compactifications for the domain and the
codomain of an affine rational map which parametrizes a hypersurface. We
show that the closure of the image of this map (with possibly some other extra
hypersurfaces) can be represented by a matrix of linear syzygies. We compactify
into an -dimensional projective arithmetically
Cohen-Macaulay subscheme of some . One particular interesting
compactification of is the toric variety associated to the
Newton polytope of the polynomials defining . We consider two different
compactifications for the codomain of : and .
In both cases we give sufficient conditions, in terms of the nature of the base
locus of the map, for getting a matrix representation of its closed image,
without involving extra hypersurfaces. This constitutes a direct generalization
of the corresponding results established in [BuseJouanolou03],
[BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and
[Botbol09].Comment: 2 images, 28 pages. To appear in Journal of Pure and Applied Algebr
Curvilinear Base Points, Local Complete Intersection and Kozsul Syzygies in Biprojective Spaces
We prove analogs of results of Cox/Schenck on the structure of certain ideals
in the bigraded polynomial ring k[s,u;t,v].Comment: 12 page
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