Let X ↪ P r X \hookrightarrow \mathbb {P}^r be a smooth projective variety defined by homogeneous polynomials of degree ≤ d \leq d . We give an explicit upper bound on the order of the torsion subgroup ( NS ⁡ X ) t o r (\operatorname {NS} X)_{\mathrm {tor}} of the Néron–Severi group of X X . The bound is derived from an explicit upper bound on the number of irreducible components of the scheme C D i v n ⁡ X \operatorname {\mathbf {CDiv}}_n X parametrizing the effective Cartier divisors of degree n n on X X . We also give an upper bound on the number of generators of ( NS ⁡ X ) [ ℓ ∞ ] (\operatorname {NS} X)[\ell ^\infty ] uniform as ℓ ≠ c h a r k \ell \neq \mathrm {char} k varies.