4.3 Strict transforms and multiplicities
Suppose \(\pi :\widetilde{X}\to X\) is a monoidal transformation and \(C\) is an effective divisor in \(X\). The strict tranfrom of \(C\) is the closure in \(\widetilde{X}\) of \(\pi ^{-1}(C\cap (X\setminus P))\). It is also the closed subscheme of \(\widetilde{X}\) obtained by blowing up \(P\) in \(C\). We get it from \(\pi ^* C\) by throwing away the exceptional divisor \(E\). People write it as \(\pi _*^{-1} C\) or as \(\widetilde{C}\).
Suppose \(C\) is as above and \(f\) is the local equation for \(C\) at \(P\). The multiplicity of \(C\) at \(P\) is the number \(\mu _P(C)\) defined to be the largest integer \(r\) such that \(f\in \mathfrak {m}_P^r\).
Let \(C\) be an effective divisor on \(X\) and let \(P\) be a point of multiplicity \(r\) on \(C\). Let \(\pi :\widetilde{X}\to X\) be the monoidal transformation with center \(P\). Then
where \(E\) is the exceptional divisor.
This is [ 4 , Propsition V.3.6 ] .
Let’s also do this in the basic example of the blow up of the origin on \(\mathbb {A}^2\). Then \(C\) is given by a polynomial equation \(f(x,y) = 0\), with \(f = \sum _{i\geq r} f_i\) where \(f_i\) is homogeneous of degree \(i\). The divisor \(\pi ^* C\) is given by the equation \(f(\pi (z)) = 0\). But it’s easier to look at it on the open sets \(U = \operatorname{\mathrm{Spec}}k[u,y]\) and \(V = \operatorname{\mathrm{Spec}}k[x,v]\) where \(u=1\) and \(v=1\) respectively.
The map \(\pi |U:U\to \mathbb {A}^2\) is given by \(\pi (u,y) = (uy,y)\). So \(f_i(\pi (z)) = y^i f_i(u,1)\). Therefore, \(\pi ^*C\cap U\) is the curve define by \(y^r \tilde{f}\), where \(\tilde{f} = \sum _{i\geq r} y^{i-r}f_i(u,1)\). The polynomial \(f_r(u,1)\) is nonzero (because \(f\) is nonzero), and \(V(y) = E\). So \(\pi ^*C = rE + V(\tilde{f})\). But \(V(\tilde{f})\setminus E = \widetilde{C}\setminus E\). So the result follows.
In the setting of Proposition 4.15, we have
\(\widetilde{C} \cdot E = r\);
\(\widetilde{C}^2 = C^2 - r\).