Complex Analysis and Geometry by Daniel Barlet (auth.), Vincenzo Ancona, Alessandro Silva

By Daniel Barlet (auth.), Vincenzo Ancona, Alessandro Silva (eds.)

The papers during this wide-ranging assortment document at the result of investigations from a couple of associated disciplines, together with complicated algebraic geometry, complicated analytic geometry of manifolds and areas, and complicated differential geometry.

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We know that HP is b-complete, and this gives b'"HP c HP. Therefore, HP is a q[b]]-module of finite type, and HP is a regular (a, b)-module. Then via 4J we get that 4J(HP) is b-Ia' stable in EP[b- l ] of finite type over q[b]] and contains P. Since b-Ia' = b-Ia + 1, we conclude that EP is a regular (a, b)-module. 0 We conclude with aremark that shows that the (a, b)-module point of view is relevant to our understanding of the way in which the raots of the Bernstein polynomial jump in a p-constant family of isolated singularities of hypersurfaces.

1). Nevertheless, this weaker condition is interesting, and we shalllook at it now. 2. We shall say that an (a, b)-module Eis loeal iff there N * such that a'"E c bE. 1. Let E = IC[[ b]]el EB IC[[ b]]e2 and define EXAMPLE and Then we have a2el = bel and a2e2 = b(a + b)el so that a2EcbE (note that abEcbE is always true for an (a, b)-module). Let us show that E is not regular. By induetion on n E N we want to prove that, for any p E [0, n], b-P el and b-P e2 are in (b-Ia)jE. This is clear for n = o.

Camplex Analysis and Geametry, edited by Vincenzo Ancona and Alessandro Silva. Plenum Press, New York, 1993. , Spec(lR))) has property P and if the regular locus Reg(S) is dense in S, then Jor a dense set oJ parameters SES, property P holds Jor J in the points oJ the fiber J- 1(s). , the theorem ofFrisch on generic flatness). 2. We also show that properties ofthe tangent cones of the local rings can be treated in a satisfactory way; cf. 3. To show the efficiency of these criteria, we verify them for a long list of examples; see Section I.

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