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I used divisors because they are the first example of something alway gorenstein I had... Now I understand! I was pretty sure that there were something not right in how I understood the proposition but I could not see what point I was missing. Thank you very much
Thank you very much. As a matter of fact I think I managed: infact my 27 polynomials, even if scary-looking in the whole, turned out to be quite simple. For example, for many of them both $F$ and $G$ are of prime degree, and for the others, they have a degree written as the product of just to prime... this allowed me to esclude at once many decompositions...
Wow, that is really interesting. I do not know how easy it will be to apply this criteria (I have to check the irreducibility of 27 polynomial over the complex numbers and I am reclutant to use a computer algebra system since over the complex number they use approximations and therefore their answer cannot really considered as a PROOF of irreducibility...:D) but I will try anyway. Thank you very much for your very useful and detailed anwer, best
Thank you very much both do Hailong and Karl. It seems to me that many results holds just for subvarieties and not for subschemes. Am I correct? I have a smooth varieties of dimension 5, $X$ and two subschemes $B$ (reduced irred of dimension 3 and $Z$ (pure of dimension 3 (but it could have embedded components of lower dimension) and it has at least 2 non-embedd copts and one of them does not intersect $B$). They intersect properly. The higher tor's of their strct sheaves vanish. It seems to me that could not somewhat be true? If it where true, would it tell me that both $Z$ and $B$ are CM?
