Lemma 10.8.10EB08BA5202E3

Proof. We argue by induction on the number of blowups. On $Z$ itself, $A_ E=b_ E=1$, and every other prime divisor $D$ of $Z$ has $A_ D=1$ and $b_ D=0$. Strict transforms keep their log discrepancies and orders, hence their margins. Since every exceptional divisor lies in the boundary, a prime divisor of a model outside the boundary is the strict transform of a prime divisor of $Z$ other than $E$, so it has $A_ D=1$ and $b_ D=0$. Suppose that the statement holds on a model. Blowing up a center of codimension one is an isomorphism and creates no new divisor; it only adds the center to the boundary, and the center then has margin one if it was not a boundary component. It therefore suffices to show that the exceptional divisor $E_{\mathrm{new}}$ of the blowup of a center $C$ of codimension $c\geq 2$ has a positive margin.

Suppose that $C$ is contained in exactly $r$ boundary components, near its generic point. Since $C$ has normal crossings with the boundary, $r\leq c$. Near such a point choose coordinates $(z,w)$, $w=(w_1,\ldots ,w_ c)$, in which $C=\{ w=0\} $ and the boundary components through the point are coordinate hyperplanes; those containing $C$ are then hyperplanes $\{ w_ j=0\} $. By proposition 9.6.3, in the coordinates $(z,t,v)$ of (9.6.4) we have $E_{\mathrm{new}}=\{ t=0\} $, and the Jacobian determinant of the blowup is $\pm t^{c-1}$, so it vanishes to order $c-1$ along $E_{\mathrm{new}}$. A local equation of a component $\{ w_ j=0\} $ containing $C$ is a unit times $w_ j$ (lemma 6.12.1(1)), and $w_ j$ pulls back to $t$ or to $tv_ j$, which vanishes to order one along $E_{\mathrm{new}}$, because $v_ j$ does not vanish identically on $\{ t=0\} $ (lemma 6.12.1(2),(4)). Thus the pullback of a local equation of each component containing $C$ vanishes to order one along $E_{\mathrm{new}}$. Hence $A_{\mathrm{new}}=c+\sum _{D\supset C}(A_ D-1)$ and $b_{\mathrm{new}}=\sum _{D\supset C}b_ D$, the sums running over the boundary components containing $C$ (other divisors through $C$ have $A_ D=1$ and $b_ D=0$ and contribute nothing). Therefore

All terms on the right are nonnegative by the induction hypothesis. If one of the components containing $C$ has a positive margin, so does $E_{\mathrm{new}}$. Otherwise only the strict transform of $E$ can contain $C$, so $r\leq 1$ and $c-r\geq 1$, and again the margin of $E_{\mathrm{new}}$ is positive. $\square$