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A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed.

Some theorems on normality:

  1. A local ring of dimension 1 is normal if and only if it is regular.

  2. (Serre's criterion) A scheme is normal if and only if it is nonsingular in codimension 0 and codimension 1 and every stalk at a generic point of an irreducible closed subset with dimension $\ge 2$ has depth at least 2.

  3. Every rational function on a normal scheme with no poles of codimension 1 is regular.

  4. (Zariski connectedness): If $f:X\rightarrow Y$ is a proper birational map of noetherian integral schemes and $Y$ is normal, then every fiber is connected.

  5. Normal schemes over $C$ are topologically unibranched.

But the proofs I've seen are fairly ad-hoc, and I was wondering if there's some geometric perspective that would clarify these results. The only result here thats an "iff" is Serre's criterion, but I don't understand depth geometrically so I'm not sure how to interpret it.

Is there some nice geometric perspective on normality?

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    In (4), suppose $f$ is proper and birational.2012-10-02
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    Sorry, I have edited the post.2012-10-02
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    I don't have the "Red book on varieties and schemes" of Mumford under hands, but I remember there is a page on normal varieties.2012-10-02
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    Shouldn't Serre's criterion just be R1 and S2? You have "nonsingular in codimension 0" in there which means the whole thing is nonsingular? R1=regular in codimension 1 and S2=depth condition you wrote.2012-10-03
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    @Matt: points of codimension 0 are the generic points of $X$. So regular in codimension $0$ meance $X$ (when noetherian) contains a reduced dense open subscheme. Condition $(R_1)$ means regular in codimension $\le 1$.2012-10-03
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    Careful, Serre's criterion only works for locally Noetherian schemes. It is far from true in general.2012-11-29

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