  We consider bond percolation on $\Z^d\times \Z^s$ where edges of $\Z^d$ are
open with probability $p<p_c(\Z^d)$ and edges of $\Z^s$ are open with
probability $q$, independently of all others. We obtain bounds for the critical
curve in $(p, q)$, with $p$ close to the critical threshold $p_c(\Z^d)$. The
results are related to the so-called dimensional crossover from $\Z^d$ to
$\Z^{d+s}$.
