 Hyperplane section

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.