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Improvement of Grauert-Riemenschneider's theorem for a normal surface

Jean Giraud (1982)

Annales de l'institut Fourier

Let X ˜ be a desingularization of a normal surface X . The group Pic ( X ˜ ) is provided with an order relation L ≫ _ 0 , defined by L . V ≤ 0 for any effective exceptional divisor V . Comparing to the usual order relation we define the ceiling of L which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...

Intersection of analytic curves

Tadeusz Krasiński, Krzysztof Jan Nowak (2003)

Annales Polonici Mathematici

We give a relation between two theories of improper intersections, of Tworzewski and of Stückrad-Vogel, for the case of algebraic curves. Given two arbitrary quasiprojective curves V₁,V₂, the intersection cycle V₁ ∙ V₂ in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V₁,V₂). We also give short proofs of two known effective formulae for the intersection cycle V₁ ∙ V₂ in terms of local parametrizations of the curves.

Intersection theory and separation exponent in complex analytic geometry

Ewa Cygan (1998)

Annales Polonici Mathematici

We consider the intersection multiplicity of analytic sets in the general situation. We prove that it is a regular separation exponent for complex analytic sets and so it estimates the Łojasiewicz exponent. We also give some geometric properties of proper projections of analytic sets.

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