# zbMATH — the first resource for mathematics

Relations between numerical data of an embedded resolution. (English) Zbl 0742.14009
The author considers the embedded resolution $$h:X\to X_ 0$$ of the singularities of a hypersurface $$Y$$ in the affine space $$X=\mathbb{A}^{n+1}$$. Let $$Y_ i^{(r)}$$, $$i\in I$$, be the strict transforms of the irreducible components of $$Y$$ and $$E_ i^{(r)}$$, $$1\leq i\leq r$$, be the irreducible components of the exceptional divisor, then $$\left(\bigcup_{i\in I}Y_ i^{(r)}\right)\cup\left(\bigcup^ r_{i=1}E_ i^{(r)}\right)$$ is a normal crossing divisor on $$X$$.
The numerical data $$(N_ i,\nu_ i)$$ are defined by: $$h^{- 1}(Y)=\sum_{i\in I}N_ iY_ i^{(r)}+\sum^ r_{i=1}N_ iE_ i^{(r)}$$ and $$K_ X=h^{-1}(K_{X_ 0})+\sum_{i\in I}(\nu_ i- 1)Y_ i^{(r)}+\sum^ r_{i=1}(\nu_ i-1)E_ i^{(r)}$$. When $$Y$$ is an irreducible plane curve there are some relations between these numbers and the author wants to generalize these results for any hypersurface $$Y\subset\mathbb{A}^{n+1}$$. He gets a relation for the canonical divisor on a divisor $$E=E_ j^{(r)}$$, $$1\leq j\leq r$$: Let $$E_ i'$$, $$i\in T$$, be the intersection $$E_ i^{(r)}\cap E$$ or $$Y_ i^{(r)}\cap E$$ of $$E$$ with another component of $$h^{-1}(Y)$$, then $$N_ jK_ E=\sum_{i\in T}((\nu_ i-1)N_ j-\nu_ jN_ i)E_ i'$$ in $$\hbox{Pic}(E)$$. — For a fixed $$E_ j^{(r)}$$, he gets also some relations between the numerical data corresponding to the irreducible components $$E_ i^{(r)}$$ which intersect $$E_ j^{(r)}$$ and which appear “before $$E_ j^{(r)}$$ in the resolution process”. To get these relations he needs to look at the succession of blowing-up $$g_ i:X_{i+1}\to X_ i$$ with non-singular center $$D_ i$$ such that the map $$X=X_ r\to X_{r-1}\to\cdots\to X_ 0$$ is the embedded resolution. Let $$h_ j$$ be the composed map $$h_ j:X_ j\to X_ 0$$. Let $$E_ j^{(r)}$$ be the strict transform on $$X$$ of the exceptional divisor $$E=E_ j^{(j)}$$ of $$g_{j-1}:X_ j\to X_{j-1}$$, i.e. $$E=g^{-1}_{j-1}(D)$$ with $$D=D_{j-1}$$, $$\Pi=g_{j-1\mid_ E}$$, $$k=\hbox{codim}(D,X_{j-1})$$. Let $$E_ i^{(r)}$$, $$i\in T$$, $$T\subset\{1,\ldots,r\}\cup I$$, be the irreducible components of $$h^{- 1}(Y)$$ such that the $$E_ i'=E_ i^{(r)}\cap E_ j^{(r)}$$ are the strict transforms in $$E_ j^{(r)}$$ of the irreducible components of $$E\cap(h_ j^{-1}(Y)\backslash E)$$, let $$\alpha_ i=(\nu_ i- (\nu/N)N_ i)$$; then $$\sum_{i\in T}d_ i(\alpha_ i-1)+k=0$$, where $$d_ i$$ is the degree of the cycle $$E_ i'.F$$ on the general fibre $$F=\mathbb{P}^{k-1}$$ of $$\Pi:E\to D$$. If $$d_ i=0$$ there exists a divisor $$B_ i$$ on $$D$$ such that $$E_ i'=g^{-1}_{j-1}(B_ i)$$, and in $$\hbox{Pic}(D)$$: $$\sum_{i\in T,d_ i\neq 0}{1\over kd_ i^{k- 1}}(\alpha_ i-1)\Pi_ *(E_ i'{}^ k)+\sum_{i\in T,d_ i=0}(\alpha_ i-1)B_ i=K_ D$$.
Reviewer: M.Vaquie (Paris)

##### MSC:
 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J17 Singularities of surfaces or higher-dimensional varieties
Full Text: