Degree of a projective variety

Remarks

Equivalently, $\deg X$ is the number of points of $X \cap L$ for a general linear subspace $L \subseteq \mathbb{P}^n$ of dimension $n - d$.

Example: let $X = V(f) \subseteq \mathbb{P}^2$ with $f$ an irreducible homogeneous polynomial of degree $e$. The exact sequence $0 \to S(-e) \to S \to S/(f) \to 0$ gives $P_X(t) = \binom{t+2}{2} - \binom{t-e+2}{2} = et - e(e-3)/2$. Here $d = 1$ and the leading coefficient is $e$, so $\deg X = e$.

Non-example of the geometric form: the conic $X = V(xz - y^2)$ has degree $2$. The line $y = 0$ meets it in the two points $[1:0:0]$ and $[0:0:1]$. The tangent line $x = 0$ meets it only in $[0:0:1]$. The tangent line is not general, so it does not compute the degree; it counts the point with multiplicity $2$.

The degree depends on the embedding: $\mathbb{P}^1$ embedded as a line has degree $1$, and embedded as the conic above it has degree $2$. For a hypersurface $V(f)$ with $f$ irreducible, the degree of the variety is the degree of $f$.