Lemma 43.16.3. Let $X$ be a nonsingular variety. Let $W,V \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ with generic point $\xi $. Suppose the ideal of $V$ in $\mathcal{O}_{X, \xi }$ is cut out by a regular sequence $f_1, \ldots , f_ c \in \mathcal{O}_{X, \xi }$. Then $e(X, V\cdot W, Z)$ is equal to $c!$ times the leading coefficient in the Hilbert polynomial
\[ t \mapsto \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{W, \xi }/(f_1, \ldots , f_ c)^ t,\quad t \gg 0. \]
In particular, this coefficient is $> 0$.
Proof.
The equality
\[ e(X, V\cdot W, Z) = e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c) \]
follows from the more general Lemma 43.16.2. To see that $e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c)$ is $> 0$ or equivalently that $e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c)$ is the leading coefficient of the Hilbert polynomial it suffices to show that the dimension of $\mathcal{O}_{W, \xi }$ is $c$, because the degree of the Hilbert polynomial is equal to the dimension by Algebra, Proposition 10.60.9. Say $\dim (V) = r$, $\dim (W) = s$, and $\dim (X) = n$. Then $\dim (Z) = r + s - n$ as the intersection is proper. Thus the transcendence degree of $\kappa (\xi )$ over $\mathbf{C}$ is $r + s - n$, see Algebra, Lemma 10.116.1. We have $r + c = n$ because $V$ is cut out by a regular sequence in a neighbourhood of $\xi $, see Divisors, Lemma 31.20.8 and then Lemma 43.13.2 applies (for example). Thus
\[ \dim (\mathcal{O}_{W, \xi }) = s - (r + s - n) = s - ((n - c) + s - n) = c \]
the first equality by Algebra, Lemma 10.116.3.
$\square$
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