  Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined
by homogeneous polynomials of degree $\leq d$. We give explicit upper bounds on
the order of the torsion subgroup $(\mathrm{NS} \, X)_{\mathrm{tor}}$ of the
N\'eron-Severi group of $X$. The bounds are derived from an explicit upper
bound on the number of irreducible components of either the Hilbert scheme
$\mathbf{Hilb}_Q X$ or the scheme $\mathbf{CDiv}_n X $parametrizing the
effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on
the number of generators of $(\mathrm{NS} \, X)[\ell^\infty]$ uniform as $\ell
\neq \mathrm{char}\, k$ varies.
