  The next generation of large sky photometric surveys will finally be able to
use arc statistics as a cosmological probe. Here we present the first of a
series of papers on this topic. In particular, we study how arc counts are
sensitive to the variation of two cosmological parameters: the (total) matter
density parameter, $\Omega_m$, and the normalisation of the primordial power
spectrum, expressed in terms of $\sigma_8$. Both these parameters influence the
abundances of collapsed structures and their internal structure. We compute the
expected number of gravitational arcs with various length-to-width ratios in
mock light cones, by varying these cosmological parameters in the ranges
$0.1\leq\Omega_m\leq0.5$ and $0.6\leq\sigma_8\leq 1$. We find that the arc
counts dependence on $\Omega_m$ and $\sigma_8$ is similar, but not identical,
to that of the halo counts. We investigate how the precision of the constraints
on the cosmological parameters based on arc counts depends on the survey area.
We find that the constraining power of arc statistics degrades critically only
for surveys covering an area smaller than $10\%$ of the whole sky. Finally, we
consider the case in which the search for arcs is done only in frames where
galaxy clusters have been previously identified. Adopting the selection
function for galaxy clusters expected to be detected from photometric data in
future wide surveys, we find that less than $10\%$ of the arcs will be missed,
with only a small degradation of the corresponding cosmological constraints.
