Hi,
if we could write a classification about the known regularity which is the known class of schemes that are immediately less good than normal schemes? And which properties have they?
thank you



less than normal
"Immediately less good than normal schemes" probably does not make much sense, since "goodness" depends on what you want to do.


For instance, often one likes normal varieties since it is possible to define on them a canonical (Weil) divisor $K$. This does not depend really on normality, but only on the fact that any normal variety $X$ has a singular locus of codimension at least $2$: one considers the canonical divisor on the smooth locus $X^{0}$and then push it forward on $X$.


Thus, one of the (many) possible answers to your question could be "the class of schemes whose singular locus has codimension at least $2$". 


The two conditions "normal" and "singular locus of codimension at least $2$" are equivalent for hypersurfaces of $\mathbb{P}^n$, but in general the second is weaker.


For instence, the union of two $2$-planes in $\mathbb{P}^4$ intersecting in a single point is not normal, as follows immediately from Zariski's Main Theorem.


