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Normal varieties by Silvio Greco

By Silvio Greco

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B. Analytic branches. Let V : F = 0 be a complex hypersurface passing through the origin 0. Then F is a holomorphic function near 0, and thus it has a « u n iqu e» factorization in irreducible factors (see [47] vol. I I ) : F = F 1- . . - Fn . The (germs of) analytic hypersurfaces F t = 0 are the branches of V at 0 . 49 Normal varieties This TJ of 0 namely This means that in a sufficiently small (euclidean) neighborhood F n U is the union of n irreducible analytic hypersurfaces the F { = 0 . 4.

I f gr2 (A ) is a normal domain, then A is normal. P roof : See [26], p. D). 14. Co r o l l a r y : (a) A regular local ring is normal. (b) A simple point of a variety is normal. 6. 15. R e m a k e s : (i) Let 7 : 1 1 — Z 2= 0 . 27), but the origin is singular for 7 . 14 (b). 6. 12 (6). 12. 16: T h eorem . A regular local ring is UFD. g. [26], p. 142, Th. 48. 13. 17. P roposition : Let A be a local ring and let I c A be an ideal generated by a regular sequence. I f A f l is normal, then A is normal.

6). I t is also equivalent to say that the integral closure of 0 VfX is a local ring (in general it is semilocal). Note that a normal point is always unibranch. This, definition can be extended to subvarieties in the following way (we assume for simplicity that the characteristic of Tc is zero). 10. D efinition : Let V be a reduced variety over a field Jc of characteristic zero, and let W be an integral subvariety of V. Let p : V' - » 7 be the normalization, and put W* = [p“ i(W)]redC y [that is: if A is the ring of V and P is the prime ideal of W in A , the ring of W* is A '/r a d (P A ')] (a) W is unibranch (UB) if TF* is irreducible (hence integral) (b) W is geometrically unibranch (GUB) if W is unibranch and moreover p induces a birational morphism p : W* -> W.

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