Example 52.18.4. Let $k$ be a field and $Y = \mathbf{A}^ n_ k$. Denote $\delta : Y \to \mathbf{Z}_{\geq 0}$ the usual dimension function.

1. If $Z = \{ z\}$ for some closed point $z$, then

1. $\delta _ Z(y) = \delta (y)$ if $y \leadsto z$ and

2. $\delta _ Z(y) = \delta (y) + 1$ if $y \not\leadsto z$.

2. If $Z$ is a closed subvariety and $W = \overline{\{ y\} }$, then

1. $\delta _ Z(y) = 0$ if $W \subset Z$,

2. $\delta _ Z(y) = \dim (W) - \dim (Z)$ if $Z$ is contained in $W$,

3. $\delta _ Z(y) = 1$ if $\dim (W) \leq \dim (Z)$ and $W \not\subset Z$,

4. $\delta _ Z(y) = \dim (W) - \dim (Z) + 1$ if $\dim (W) > \dim (Z)$ and $Z \not\subset W$.

A generalization of case (1) is if $Y$ is of finite type over a field and $Z = \{ z\}$ is a closed point. Then $\delta _ Z(y) = \delta (y) + t$ where $t$ is the minimum length of a chain of curves connecting $z$ to a closed point of $\overline{\{ y\} }$.

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