This paper provides a starting point for generic quantifier
elimination by partial cylindrical algebraic decomposition
(\textsc{pcad}). On input of a first-order formula over the reals
generic \textsc{pcad} outputs a theory and a quantifier-free formula.
The theory is a set of negated equations in the free variables of the
input formula. The quantifier-free formula is equivalent to the input
for all parameter values satisfying the theory. For obtaining this
generic elimination procedure, we derive a generic projection operator
from the standard Collins--Hong projection operator. Our operator
particularly addresses formulas with many parameters thus filling a
gap in the applicability of \textsc{pcad}. It restricts decomposition
to a reasonable subset of the entire space. The above-mentioned theory
describes this subset. The approach is compatible with other
improvements in the framework of \textsc{pcad}. It turns out that the
theory contains assumptions that are easily interpretable and that are
most often non-degeneracy conditions. The applicability of our generic
elimination procedure significantly extends that of the corresponding
regular procedure. Our procedure is implemented in the computer logic
system \textsc{redlog}.