 Hyperplane divisor

For $n\ge 1$, a hyperplane divisor on projective space ${\mathbb{P}}_k^n$ is the effective Cartier divisor cut out by a nonzero homogeneous linear polynomial. Its divisor line bundle is $\mathcal{O}(1)$, and its class $H$ generatesA degree-$d$ homogeneous equation defines a projective hypersurface divisor linear equivalence of divisors to $dH$. Different projective hyperplanes are linearly equivalent, since the ratio of their defining linear equations is a rational function with divisor equal to their difference. The dimension restriction matters: ${\mathbb{P}}_k^0$ has no codimension-one points and has trivial divisor class group.