an:00013846
Zbl 0742.14009
Veys, W.
Relations between numerical data of an embedded resolution
EN
Am. J. Math. 113, No. 4, 573-592 (1991).
00157575
1991
j
14E15 14J17
embedded resolution of the singularities of a hypersurface; normal crossing divisor; blowing up; exceptional divisor
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 \(\hbox{Pic}(D)\): \(\sum_{i\in T,d_ i\neq 0}{1\over kd_ i^{k- 1}}(\alpha_ i-1)\Pi_ *(E_ i'{}^ k)+\sum_{i\in T,d_ i=0}(\alpha_ i-1)B_ i=K_ D\).
M.Vaquie (Paris)