I am not an expert in the geometric meaning of normality/Cohen Macaulay, so the following questions could seem very stupid.
- Are there examples of connected varieties over a field whose irreducible components are smooth and they intersect in a closed of codimension $> 1$?
- Are these varieties normal even if they are not irreducible?
- If the whole variety is Cohen-Macaulay, with smooth irreducible components, does this imply that they intersect in codimension $> 1$?
- 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?
- 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.