Thanks! Well I will stick to rational singularities then :'(!!!! Again thank you

First of all, thank you very much. To answer your question, I need to find out some hypothesis that do not allow these loci to be too big. In particular I want that for every positive $i$ the codimension in $A$ of $R^if_*(\mathcal{O}_Y$ is greater than $i+2$. If $D$ has rational singularities I am ok, but this hypo is somewhat too strong. Certainly $D$ has to be normal. but I really do not know how to manage $i>1$... thank again for your help best regards

Thank you very much for your answer! I will check the reference about cohen maculay rings that you suggested! The fact is that the subscheme $Z$ I am working with is very simple: set theoretically is a union of very nice smooth varieties (i.e complex tori). The intersection points between different components are not a big deal, what I would like to check is that the set-theoretic description holds also scheme-theoretically (or that at least one of the complex tori is not "doubled"). thak you again for tuor time!

Thank you very much for your time! this was really helpfull.

Thank you very much!!! I believed it was true but I could not find the right reference!!! best wishes!

I see... So, as I feared, I did get it completely wrong :-(... thank you again and sorry for taking so much of your time! best regards