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A depth formula for generic singularities and their weak normality. (English) Zbl 1133.14003

Let \(X\) be an \(r\)–dimensional smooth projective variety and \(\pi:X\to \mathbb P^m\) a generic projection, \(r+1\leq m\leq 2r\). It is proved that at any generic point of \(\pi(X)\) of multiplicity \(\mu\) the depth of the local ring is \(r-(\mu-1)(m-r-1)\).

MSC:

14B05 Singularities in algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
13F45 Seminormal rings
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