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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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