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 V 1 , V 2 , the intersection cycle V 1 •V 2 in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V 1 , V 2 ). We also give short proofs of two known effective formulae for the intersection cycle V 1 •V 2 in terms of local parametrizations of the curves. Introduction. For two arbitrary
more » ... ly dimensional analytic subsets V 1 , V 2 (or, in general, for two analytic cycles V 1 , V 2 ) of a complex manifold M , Tworzewski [T] defined an intersection product V 1 • V 2 which is an analytic cycle. This theory was initiated in the case of improper isolated intersections by Achilles, Winiarski and Tworzewski [ATW], who made use of Draper's ideas (cf. [D]) concerning proper intersections in complex analytic geometry. The intersection cycle V 1 • V 2 coincides with the classical one for every proper intersection of V 1 and V 2 . On the other hand, for two arbitrary purely dimensional quasiprojective varieties V 1 , V 2 over a field K we have the Vogel intersection cycle v(V 1 , V 2 ) (see [CFV, G1, G2] ), which is an algebraic cycle defined over a pure transcendental extension of K. So, we may distinguish in v(V 1 , V 2 ) its rational part v rat (V 1 , V 2 ) (i.e. the part defined over K) and transcendental part v tr (V 1 , V 2 ): v(V 1 , V 2 ) = v rat (V 1 , V 2 ) + v tr (V 1 , V 2 ).
doi:10.4064/ap80-0-16 fatcat:4pjvih2j7jgqjemchfqormov6a