  We compute the Picard group of the moduli stack of smooth curves of genus $g$
for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base
our proof on the computation of some relations in the integral Chow ring of
certain moduli stacks of smooth complete intersections. As a byproduct, we
compute the cycle classes of some divisors on $\mathcal{M}_g$.
