Minimal Generators for the Rees Algebra of Rational Space Curves of Type (1,1,d-2)

We provide an algorithm to find a minimal set of generators for the Rees algebra associated to  rational space curves of type (1,1,d-2) in projective 3-space based solely  on a $\mu$-basis of the curve. We also illustrate the geometry behind the generators via a case study of rational quartic space curves.

[1]  A. Simis,et al.  ELIMINATION AND NONLINEAR EQUATIONS OF REES ALGEBRA , 2009, 0911.2569.

[2]  Falai Chen,et al.  Implicitization using moving curves and surfaces , 1995, SIGGRAPH.

[3]  Falai Chen,et al.  The moving line ideal basis of planar rational curves , 1998, Comput. Aided Geom. Des..

[4]  Ron Goldman,et al.  Implicitizing Rational Curves by the Method of Moving Algebraic Curves , 1997, J. Symb. Comput..

[5]  Laurent Busé,et al.  Implicitizing rational hypersurfaces using approximation complexes , 2003, J. Symb. Comput..

[6]  Thomas W. Sederberg,et al.  Curve implicitization using moving lines , 1994, Comput. Aided Geom. Des..

[7]  Joe Harris,et al.  Curves in projective space , 1982 .

[8]  Jooyoun Hong,et al.  On the homology of two-dimensional elimination , 2007, J. Symb. Comput..

[9]  Robin Hartshorne,et al.  Algebraic geometry , 1977, Graduate texts in mathematics.

[10]  Ron Goldman,et al.  Set-theoretic generators of rational space curves , 2010, J. Symb. Comput..

[11]  J. William Hoffman,et al.  Syzygies and the Rees algebra , 2008 .

[12]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[13]  J. Herzog,et al.  Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces , 1997 .

[14]  A. Gimigliano,et al.  Projectively Normal but Superabundant Embeddings of Rational Surfaces in Projective Space , 1994 .

[15]  Laurent Busé,et al.  On the equations of the moving curve ideal , 2007, ArXiv.

[16]  N. Trung,et al.  The diagonal subalgebra of a blow-up algebra , 1998 .

[17]  Ron Goldman,et al.  Axial moving planes and singularities of rational space curves , 2009, Comput. Aided Geom. Des..

[18]  W. Vasconcelos Arithmetic of Blowup Algebras , 1994 .

[19]  Laurent Busé,et al.  ON THE CLOSED IMAGE OF A RATIONAL MAP AND THE IMPLICITIZATION PROBLEM , 2002, math/0210096.

[20]  Ron Goldman,et al.  Mu-bases for Polynomial Systems in One Variable , 2009, Comput. Aided Geom. Des..

[21]  A. Gimigliano,et al.  On the Ideal of Veronesean Surfaces , 1993, Canadian Journal of Mathematics.

[22]  S. X. Descamps Scrolls and quartics , 1982 .

[23]  David A. Cox The moving curve ideal and the Rees algebra , 2008, Theor. Comput. Sci..

[24]  Bernd Ulrich,et al.  Rees algebras of ideals with low codimension , 1996 .