= Hyperplane divisor
{title2=$H$}
For $n\geq1$, 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 $H$ generates
$$
\operatorname{Cl}(\mathbb P_k^n)\cong\mathbb Z.
$$
A 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>.
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