  Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective
varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete
intersections), there are only finitely many components of the Hilbert scheme
parametrizing irreducible, smooth, projective, low codimensional subvarieties
$X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq \lambda
d^{\epsilon_1}+c(\sum_{1\leq h < \epsilon_2}p_g(X^{(h)})), $$ where $d$, $K_X$
and $H_X$ denote the degree, the canonical divisor and the general hyperplane
section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general
linear section of $X$ of dimension $h$, and where $\lambda$, $\epsilon_1$ and
$\epsilon_2$ are suitable positive real numbers depending only on the dimension
of $X$, on $a$ and on the ambient variety $V$. In particular, except for
finitely many families of varieties, the canonical map of any irreducible,
smooth, projective, low codimensional subvariety $X$ of $V$, is birational.
