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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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    $\begingroup$ 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 ) . $\endgroup$ Apr 3, 2012 at 20:32
  • $\begingroup$ Also, you might tell us what the motivation for all these questions is and what you have tried in order to solve them. $\endgroup$ Apr 3, 2012 at 21:38
  • $\begingroup$ 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. $\endgroup$
    – Harry
    Apr 4, 2012 at 15:17
  • $\begingroup$ Dear ulla, Are you still interested in this question? Regards, $\endgroup$
    – Matt E
    May 4, 2012 at 15:56

1 Answer 1

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Consider $A=k[x,y]/(xy)$ and $X=\mathrm{Spec} \ A$. This is a connected variety with two irreducible components that meet in a closed subscheme of codimension $1$. (It's the union of the $x$ and $y$-axis in the affine place.) So that's not what you want in question 1.

Let $X$ be a connected variety with irreducible components $X_1,\ldots, X_n$.

If $X$ is not equidimensional funny things can happen. For example, You can glue the affine line to the affine plane in one point. This is a connected variety of "dimension two". Its irreducible components intersect in a point. So this answers your first question, albeit in an unsatisfactory way because your varieties are probably equidimensional. (So the answer to q1 is yes.)

For your second question you can do similar things. Just glue a singular curve to a singular surface along a smooth point. The resulting scheme is non-normal, but its irreducible components intersect in a point of codimension 2.

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