Behavior of the Fiber and the Base Points of Parametrizations Under Projections

Given a rational parametrization ${{mathcal{P}(overline {t}), {overline {t}}=(t_1, ldots, t_r)}}$ , of an r-dimensional unirational variety, we analyze the behavior of the variety of the base points of ${mathcal{P}(overline {t})}$ in connection to its generic fibre, when successively eliminating the parameters t _i . 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 t ₁-general position and radical.