Hyperplane divisor ($H$)

A hyperplane divisor on projective space is the effective Cartier divisor cut out by one nonzero linear homogeneous polynomial. Its class generates $\mathrm{Cl}({\mathbb{P}}_k^n)\cong \mathbb{Z}$.

A hyperplane section of a projective variety $X\subset {\mathbb{P}}^n$ is the scheme-theoretic intersection $X\cap H$ with a projective hyperplane $H$. Its associated line bundle is ${\mathcal{O}}_X(1)$.

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 $\mathrm{deg}[X\cap H]=(\mathrm{deg}X)(\mathrm{deg}H)=\mathrm{deg}X$.