  Let W be a projective variety of dimension n+1, L a free line bundle on W, X
in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by
sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map
from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X
in $H^0(L^d)$, f deforms in a family such that the corresponding deformations
of $Y^r$ dominate $W^r$. Under these hypotheses we give a lower bound for the
dimension of a certain linear system on the Cartesian product $Y^r$ having
certain vanishing order on a diagonal locus as well as on a double point locus.
This yields as one application a lower bound on the dimension of the linear
system |K_{Y} - (d - n + m)f^*L - f^*K_{W}| which generalizes results of Ein
and Xu (and in weaker form, Voisin). As another perhaps more surprising
application, we conclude a lower bound on the number of quadrics containing
certain projective images of Y.
