Hyperplane section

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 ${\mathcal{O}}_X(1)$.

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