less than normal

Question

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

Answer 1

"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.

Answer 2

There are several decent candidates (including the bounds on the dimension of the singular locus as mentioned above by Francesco Polizzi). Here are some others that are somewhat common.

1. Seminormal and Weakly Normal. A reduced scheme $X$ is seminormal (resp. weakly normal) if for every finite map $f : Y \to X$ satisfying:

(a) $Y$ is reduced.

(b) $f$ induces a bijection on points (closed or not)

(c) $f$ induces isomorphisms of residue fields at each point) (respectively induces a purely inseparable extension of residue fields)

... satisfying those 3 conditions is automatically an isomorphism. For example, a node is seminormal but a cusp is not. Three lines through the origin in $\mathbb{A}^2$ are not seminormal. Three coordinate axes through the origin in $\mathbb{A}^3$ is seminormal. If you don't want to change the points of your variety, seminormalization will make it as nice as you can without messing with those points.

2. G1 and S2. Normality is the same as being regular in codimension 1 (R1) and satisfying Serre's second condition (S2). A weaker version of this is G1 (Gorenstein in codimension 1) and S2. The advantage of this is that the canonical module $\omega$ still acts like a divisor on a normal scheme. The cusp and the node are G1 + S2, but three coordinate lines in $\mathbb{A}^3$ is not.

3. Seminormal + G1 + S2 (you may as well include equidimensional too). Combine the best of both worlds...