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Congruences for numerical data of an embedded resolution. (English) Zbl 0764.14008
The author considers the embedded resolution $$h:X\to X_ 0$$ of the singularities of an 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 $$(\bigcup_{i\in I}Y_ i^{(r)})\cup(\bigcup^ r_{i=1}E_ i^{(r)})$$ is a normal crossings 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 $$K_ X=h^{- 1}(K_{X_ 0})+\sum_{i\in I}(\nu_ i-1)Y_ i^{(r)}+\sum^ r_{i=1}(\nu_ i-1)E_ i^{(r)}$$.
Fix one exceptional curve $$E$$ with numerical data $$(N,\nu)$$. Let $$E_ j^{(r)}$$ be an exceptional divisor, and let $$E_ i^{(r)}$$, $$i\in T$$, $$T=I\cup\{1,\dots,r\}$$, be the components of $$h^{-1}(Y)$$ which intersect $$E$$ and appear “before $$E_ j^{(r)}$$” in the resolution process then the author gives some congruence relations between the numerical data $$(N_ i)$$ and $$(\nu_ i)$$, $$i\in T$$, which generalize the relations when $$Y$$ is a plane curve to any arbitrary $$Y\subset A^{n+1}$$. — To get these relations the author looks at the succession of blowing-ups $$g_ i:X_{i+1}\to X_ i$$ with non-singular center $$D_ i$$ such that the map $$X=X_ r\to X_{r-1}\to\dots\to X_ 0$$ is the embedded resolution. More precisely, he considers the strict transforms $$E_ j^{(i)}$$ of the exceptional divisor $$E=E_ j^{(j)}$$ of $$g_{j-1}$$, i.e. $$E=g^{-1}_{j-1}(D)$$ with $$D=D_{j-1}$$, and the blowing-up $$E_ j^{(i+1)}\to E_ j^{(i)}$$.
Let $$E_ i^{(r)}$$, $$i\in T$$, be the irreducible components of $$h^{- 1}(Y)$$ such that $${\mathcal E}_ i=E_ i^{(j)}\cap E$$ are the irreducible components of $$E\cap(h^{-1}(Y)\setminus E_ j)$$, and let $$d_ i$$ be the degree of the cycle $${\mathcal E}_ i\cdot F$$ on the general fibre $$F=\mathbb{P}^{k-1}$$ of $$\Pi=g_{j-1_{| E}}$$: $$E\to D$$, with $$k=n+1- \dim D$$. Then we get the congruences:
(B1) $$\sum_{i\in T}d_ iN_ i\equiv 0\pmod N$$.
(B$$1'$$) $$\sum_{i\in T}d_ i(\nu_ i-1)+k\equiv 0\pmod\nu$$.
If $$d_ i=0$$ there exists a divisor $$B_ i$$ on $$D$$ such that $${\mathcal E}_ i=\Pi^{-1}(B_ i)$$, and we get:
(B2) $$\sum_{i\in T,d_ i\neq 0}N_ i\Pi_ *({\mathcal E}_ i^ k)/d_ i^{k-1}+\sum_{i\in T,d_ i=0}N_ iB_ i=0$$ in $$\text{Pic} D/N \text{Pic} D$$.
(B$$2'$$) $$\sum_{i\in T,d_ i\neq 0}(\nu_ i-1)\Pi_ *({\mathcal E}_ i^ k)/d_ i^{k-1}+\sum_{i\in T,d_ i=0}(\nu_ i-1)B_ i-kK_ D=0$$ in $$\text{Pic} D/\nu \text{Pic} D$$.
If $$\mu_ t$$ is the multiplicity of the generic point of $$D_ i$$ on the strict transform $${\mathcal E}_ t^{(i)}$$ of $${\mathcal E}_ t$$ on $$E_ j^{(i)}$$, $$j\leq i<r$$, the author gets also the congruences:
(A) $$N_{i+1}\equiv\sum_{t\in T\cup\{1,\dots,i\}}\mu_ tN_ t\pmod N$$
(A$$'$$) $$\nu_{i+1}\equiv\sum_{t\in T\cup\{1,\dots,i\}}\mu_ t(\nu_ t-1)+(k-1)\pmod\nu$$.
Reviewer: M.Vaquie (Paris)

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