Hyperplane divisor ($H$)

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.

For a projective variety $X\subseteq {\mathbb{P}}^n$ and a projective hyperplane $H$, the hyperplane section is the scheme-theoretic intersection $X\cap H$. The defining linear equation restricts to a global section of the line bundle ${\mathcal{O}}_X(1)$.

To view its zero scheme as an effective Cartier divisor, the restricted section must be a non-zero-divisor locally. For an integral $X$, this holds when $H$ does not contain $X$. If $X\subseteq H$, the restriction is the zero global section and the intersection is all of $X$, rather than a codimension-one Cartier divisor. More generally one must avoid the associated points that would make the local equation a zero divisor.

For a projective plane curve not contained in a line $H$, intersection multiplicities record tangencies in the section. The hyperplane-section divisor of a projective plane curve has total degree equal to the degree of the curve. The restricted bundle ${\mathcal{O}}_X(1)$ exists whether or not the particular section is regular.

For a smooth integral projective variety over $\mathbb{C}$, a general hyperplane section in a projective embedding is smooth; if its dimension is at least two, the section is also irreducible. More general statements concern linear systems and their base loci. The smoothness statement is developed in the Stacks Project's Bertini section.

If a projective line $H$ does not contain the projective plane curve $X$, its hyperplane-section divisor iswhere $I_p(X,H)$ is the intersection multiplicity. The Bézout theorem gives $\mathrm{deg}[X\cap H]=(\mathrm{deg}X)(\mathrm{deg}H)=\mathrm{deg}X$.