## Read e-book online Algebroid Curves in Positive Characteristics PDF

By A. Campillo

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**Sample text**

We n e e d (Bennet, (or a graded projective and the manner. fl (M) ,, BIM(A) II5 II ~ n ProJ(nO__oMJMn+l) is induced by the natural co ~ Proj( graded n e0 @ M n=o Thus, onto to such a point (al the ,... the closed directions O' ,aN). n+l n=o M points of J7-1(0) the origin represented scheme homomorphism > through may The ring ~ M n) n::o 7"1-1(O) correspond in by is . an a f f i n e its one-one N-space, homogeneous called the and and coordinates divisor exceptional of/l. ~l 0 . O'=(a I ..... Moreover, { xl X is a regular Thus, system I~ Bl (A)~O' of M the h o m o m o r p h i s m is -1 (O) is point of x ' XN X.

8. vi) it = f is g yen irreducible n the that f(X,Y) of (X,Y) + f n+l C is p (X,Y) plane and We the T tangent given Then (X,Y) consider line Y- a X = 0, have X = Y = Y' f(X,Y) The may be Notations of the C under and hence + .... c ( Y - a X )n c,a f ,~ 0 . S i n c e n thus, if X=0 E: k and c ~ 0. -I(0) d e t e r m i n e d and the formal prime quadratic by transformation by we and = the point X v it is series , and the Y') f'(X~,Y constructed called X' + a X' = X 'n f'(X',Y f'(X' C' (resp. set is a homogeneous polynomial of degree j and j f is irreducible, f is a power of a linear form and n is transversal to C ~ n t(C) s curve f.

A point O in i the sequence of I infinitely ). 44 near we points say that of 0 i the +1 origin ' Oi+ 2' of ... ), : e(Oi+1)+ e(Oi+2)+... +e(Oi+r) of curve 0 i when C, CHAPTER HAMBURGER-NOETHER material section The purpose which will we expansions shall in expansions, the remaining is is in the there is is replace ALGEBROID to give following no the the CURVES. technical chapters, possibility The of In the obtaining first Puiseux Hamburger-Noether Puiseux EXPANSIONS. well known series. at all us that field ones, are developed in of true, and consider algebroid DISCUSSION, every algebroid characteristic We s h a l l the see curve zero how in classical a primitive [] curve may positive methods parametric over over an be an exhibited characteristic are not sufficient.