  We develop a new method that improves the efficiency of equation-by-equation
algorithms for solving polynomial systems. Our method is based on a novel
geometric construction, and reduces the total number of homotopy paths that
must be numerically continued. These improvements may be applied to the basic
algorithms of numerical algebraic geometry in the settings of both projective
and multiprojective varieties. Our computational experiments demonstrate
significant savings obtained on several benchmark systems. We also present an
extended case study on maximum likelihood estimation for rank-constrained
symmetric $n\times n$ matrices, in which multiprojective $u$-generation allows
us to complete the list of ML degrees for $n\le 6.$
