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On the intersection multiplicity of images under an etale morphism

Krzysztof Nowak (1998)

Colloquium Mathematicae

We present a formula for the intersection multiplicity of the images of two subvarieties under an etale morphism between smooth varieties over a field k. It is a generalization of Fulton's Example 8.2.5 from [3], where a strong additional assumption has been imposed. In a special case where the base field k is algebraically closed and a proper component of the intersection is a closed point, intersection multiplicity is an invariant of etale morphisms. This corresponds with analytic geometry where...

Semi-simple Carrousels and the Monodromy

David B. Massey (2006)

Annales de l’institut Fourier

Let 𝒰 be an open neighborhood of the origin in ℂ n + 1 and let f : ( 𝒰 , 0 ) → ( ℂ , 0 ) be complex analytic. Let z 0 be a generic linear form on ℂ n + 1 . If the relative polar curve Γ f , z 0 1 at the origin is irreducible and the intersection number ( Γ f , z 0 1 · V ( f ) ) 0 is prime, then there are severe restrictions on the possible degree n cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when ( Γ f , z 0 1 · V ( f ) ) 0 is not prime.

The membership problem for polynomial ideals in terms of residue currents

Mats Andersson (2006)

Annales de l’institut Fourier

We find a relation between the vanishing of a globally defined residue current on ℙ n and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.

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