Behavior of the fiber and the base points of parametrizations under projections

This is the author’s version of a work that was accepted for publication in Mathematics in Computer Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Mathematics in Computer Science Volume 7, Issue 2 (2013), Page 167-184 DOI 10.1007/s11786-013-0139-8Both authors are members of the of the Research Group ASYNACS (Ref. CCEE2011/R34).Given a rational parametrization P( t ), t = (t1, . . . , tr ), of an r-dimensional unirational variety, we analyze the behavior of the variety of the base points of P( t ) in connection to its generic fibre, when successively eliminating the parameters ti . For this purpose. we introduce a sequence of generalized resultants whose primitive and content parts contain the different components of the projected variety of the base points and the fibre. In addition, when the dimension of the base points is strictly smaller than 1 (as in the well known cases of curves and surfaces), we show that the last element in the sequence of resultants is the univariate polynomial in the corresponding Gröbner basis of the ideal associated to the fibre; assuming that the ideal is in t1-general position and radical.Ministerio de Ciencia e Innovació