Relations between numerical data of an embedded resolution. (English) Zbl 0756.14008

Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 397-403 (1991).
[For the entire collection see Zbl 0743.00011.]
Let \(k\) be an algebraically closed field, \(\text{char}(k)=0\), and \(f\in k[x,y]\), let \((X,k)\) be the canonical embedded resolution of the plane curve \(f=0\). Let \(E_ i\), \(i\in T\), be the irreducible components of \(h^{-1}(f^{-1}(\{0\}))\) (inverse image of \(f=0)\). To each \(i\in T\), associate \((N_ i,\nu_ i)\) where \(N_ i\) and \(\nu_ i\) are the multiplicity of \(E_ i\) in the divisor of \(f\circ h\) and \(h^*(dx\bigwedge dy)\) on \(X\). Fix one component \(E\) with its invariants \((N,\nu)\), denote by \(E_ 1,\dots,E_ k\) the other components of \(h^{-1}(f^{-1}(\{0\}))\) intersecting \(E\), then, you have: \((*)\;\sum^ k_{i=1}(\alpha_ i-1)+2=0\), where \(\alpha_ i=\nu_ i- {\nu\over N}N_ i\), \(1\leq i\leq k\).
In this conferences article, the author gives (without proofs) some generalizations of \((*)\) for hypersurfaces in \(\mathbb{A}_ n(k)\), \(3\leq n\). This problem is much deeper than in the case of plane curves:
(1) The components of \(h^{-1}(f^{-1}(\{0\}))\) may happen not to be linear projective spaces;
(2) If you blow-up a smooth variety inside several components of \(h^{- 1}(f^{-1}(\{0\}))\), you may happen not to separate them and you modify their Pic.
The author finds two formulas \(B_ 1\), \(B_ 2\) generalizing \((*)\) and a special one \(A\) for the components of \(h^{-1}(f^{-1}(\{0\}))\) created by blowing-ups of type 2.


14E25 Embeddings in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C22 Picard groups


Zbl 0743.00011