Degree of a projective variety

Summary

Let $X \subseteq \mathbb{P}^n$ be a projective variety of dimension $d$. Its degree is the number of points in which $X$ meets a general linear subspace of dimension $n - d$. Algebraically, it is $d!$ times the leading coefficient of the Hilbert polynomial of $X$. It extends the degree of a polynomial to varieties and is the basic quantity in Bézout's theorem.