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.

MSC:

 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