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I am not an expert in the geometric meaning of normality/Cohen Macaulay, so the following questions could seem very stupid.

  1. Are there examples of connected varieties over a field whose irreducible components are smooth and they intersect in a closed of codimension $> 1$?
  2. Are these varieties normal even if they are not irreducible?
  3. If the whole variety is Cohen-Macaulay, with smooth irreducible components, does this imply that they intersect in codimension $> 1$?
  4. In the cases where I have Cohen-Macaulay + intersection in codimension $> 1$, do I have normality of the whole stuff or I need also irreducibility?
  5. In general one needs normality to extend sections of a line bundle. Does the condition that the singular locus has codimension $> 1$ make it work also in the not irreducible case? By this I mean that I have a line bundle on these smooth components outside their intersections and I want to extend it to the whole variety, possibly in a unique way.

Thanks

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    You are violating the pigeonhole principle: 5 questions with 4 numbers . And I think there are two many pigeons- er- questions (although they are interesting ) .2012-04-03
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    Also, you might tell us what the motivation for all these questions is and what you have tried in order to solve them.2012-04-03
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    your question 3 doesn´t make much sense to me. If you have an irreducible CM scheme, then its irr components (there's only one) intersect as you want and the whole thing is CM. The scheme in question is then normal iff it is normal in codimension 1; see Corollary 8.2.24 of Liu's book on alg. geom.2012-04-04
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    Dear ulla, Are you still interested in this question? Regards,2012-05-04

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