Hyperplanarity of the equimultiple locus
Proc. Amer. Math. Soc. 87 (1983), 403-408
Primary 14B05; Secondary 13H05, 13H15, 14J17
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Abstract: It is known that the (local) equimultiple locus of a hypersurface defined over a field of characteristic zero is contained in a hyperplane (for example, the one given by the Tchirnhausen transformation: see Abhyankar's paper [A] for details). In this note (a) we show that this theorem is no longer true for varieties of dimension bigger than two in char , and (b) we give proofs of this statement in the cases of
(i) 'purely inseparable' surfaces,
(ii) double points of surfaces in char 2.
S. Abhyankar, Good points of a hypersurface, Adv. in Math.
68 (1988), no. 2, 87–256. MR 934366
Nagata, Local rings, Robert E. Krieger Publishing Co.,
Huntington, N.Y., 1975. Corrected reprint. MR 0460307
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