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Ce texte est consacré au système d’Euler de Kato, construit à partir des unités modulaires, et à son image par l’application exponentielle duale (loi de réciprocité explicite de Kato). La présentation que nous en donnons est sensiblement différente de la présentation originelle de Kato.
On majore la dimension de l’ensemble des hypersurfaces de dont l’intersection avec une variété projective intègre fixée n’est pas intègre. Les majorations obtenues sont optimales. Comme application, on construit, quand c’est possible, des hypersurfaces dont les intersections avec toutes les variétés d’une famille de variétés projectives intègres sont intègres. Le degré des hypersurfaces construites est explicite.
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