Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n \ge 1$, and let $F \in k[x_0,\dots,x_n]$ be a nonzero [homogeneous polynomial](/page/Homogeneous%20Polynomial) of degree $m \ge 1$. Let

\begin{align*} X:=\operatorname{Proj}(k[x_0,\dots,x_n]/(F)) \subset \mathbb P_k^n \end{align*}

be the scheme-theoretic hypersurface defined by $F$. Then there exists a nonempty Zariski-open subset $U$ of the Grassmannian $\operatorname{Gr}(1,\mathbb P_k^n)$ such that, for every projective line $L\in U$, the scheme-theoretic intersection $X\cap L$ is zero-dimensional and has length $m$. Consequently, if degree is computed by a general complementary linear section, then

\begin{align*} \deg X=m. \end{align*}