Global branches and parametrization (Q1821157)

Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let \(X=Spec(A)\), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime \({\mathfrak p}\) of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,\({\mathfrak p})\), correspond to the connected components of \(p^{-1}(Y)\), where \(p: X'\to X\) is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along \(U\cap Y\) and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,\({\mathfrak p})\), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X. For closely related results proved with completely different topological techniques see the paper by \textit{G. Tedeschi}, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)].