The Stacks project

Example 43.14.4. In this example we show that it is necessary to use the higher tors in the formula for the intersection multiplicities above. Let $X$ be a nonsingular variety of dimension $4$. Let $p \in X$ be a closed point. Let $V, W \subset X$ be closed subvarieties in $X$. Assume that there is an isomorphism

\[ \mathcal{O}_{X, p}^\wedge \cong \mathbf{C}[[x, y, z, w]] \]

such that the ideal of $V$ is $(xz, xw, yz, yw)$ and the ideal of $W$ is $(x - z, y - w)$. Then a computation shows that

\[ \text{length}\ \mathbf{C}[[x, y, z, w]]/ (xz, xw, yz, yw, x - z, y - w) = 3 \]

On the other hand, the multiplicity $e(X, V \cdot W, p) = 2$ as can be seen from the fact that formal locally $V$ is the union of two smooth planes $x = y = 0$ and $z = w = 0$ at $p$, each of which has intersection multiplicity $1$ with the plane $x - z = y - w = 0$ (Lemma 43.14.3). To make an actual example, take a general morphism $f : \mathbf{P}^2 \to \mathbf{P}^4$ given by $5$ homogeneous polynomials of degree $> 1$. The image $V \subset \mathbf{P}^4 = X$ will have singularities of the type described above, because there will be $p_1, p_2 \in \mathbf{P}^2$ with $f(p_1) = f(p_2)$. To find $W$ take a general plane passing through such a point.

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