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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