Algebroid Curves in Positive Characteristics by A. Campillo

Algebroid Curves in Positive Characteristics by A. Campillo

By A. Campillo

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https://arxiv. org/abs/1205. 5935

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Sample text

I 5. the fop page of By 1 page (29), us number primitive , the over , = v(x). - For : k(~x))] a curve, (a) e (F]) = 1. (b) [] integrally (c) There (d) Emb(El) (e) [] is exists ~< [ - ~ - : the ~< n . conditions: closed x C m k((x))] (and such thus that normal). v(x) =1 . = l. is r e g u l a r . equivalent. 5. RESOLUTION Let over k. maximal OF be the SINGULAR local denote by ring F of its TIES. an rreducible quotient algebroid field, and by curve m its ideal. I . 5. - quotients and [] We s h a l l Notations of ~< [ 0 [] = [] For each belonging to ( x-1 m ) the x x • F which have let the []-subalgebra x form of F -1 m be z/x the with generated by set z ~: m , x -I m .

5 . 1 2 . ), near the El). sequence eo TI-I(O) a point (the % mn/ n+1 n=o -- m "~ P r o j ( of the of C rings Proj(j) Q mn/ n+l n= o - - m ~ mn/ n+l ) n=o -- m Proj(-[ the is rings ideal obtain O is c a l l e d C, n=o natural Pro j( with which C. - then as M (A) and its desired. image by Furthermore a 42 i S p e c (I--]i ) (3) I ) Q ~ i ~ Spec(D) and the rest of the , is let O constructed , ,O. O respectively i. the We c l a i m the claim and if that for C=C' C and Xt=0 because sequences is for : x 1 = x 1 x.

1 . 1 curve that (x) 1 . x 1 in S complete. ), Proof: and is = p > 0. Let (()) E; k t t' (see representation curve over in a basis k((x)): 51 n-1 n Y + A (X) n-I Y + ... (X) ~ E: k I 0-~ then the degree if integers and one the the are curve [] under given a The i(x) are index be in made which the polynomial. 1 not equation curve A Remark (x) s inseparability the knows whether n i for we , , separablity Consequently, determines not. twisted have ~n-1 equation or (()) X curves. Puiseux transformations.

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