By A. Campillo

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**Sample text**

Ring the plane it the Now, we They curve 1[[] . A H a m b u r g e r be a Hamburger- }. x, in that basis in i t i s c o m p l e t e l y singularity classify of to { x,y say []. m defined system shall for tool is expansion basis of basis algebroid ideal parameter, we on the maximal parametric a transversal sequence Proposition be an i r r e d u c i b l e Hamburger-Noether depend give [] is determined. Hamburger-Noether shall find wilt make of elements evident a curve; so that in it the they singularities.

X N) (0) ;1 7 in . It gr and m follows (r7) that is of any type ..... 7 is 26 Take z E: m c b e as a b o v e , an an integer element the c such integer > 0 for that such xz that which (x 1 ..... n (~ m but z @ m s+] z ({ m S - m xN)cC n-1 (c+] q2-_ . L e t Denote ~; s+] ~ by s, n-I). ( z + m 1 Since - Xl ~: ~/ -q l hand, z = h(x] , .... - , z + m If ) = XZ + m x s+2 h z + m s+l is a series is Ig --ql • = (0) ~2 a transversal , ~ 1 . • of = h(x 1 , ' ' ' , x N ) (~ ( X l s+l = 0. -- implies where s+1 z+m We s+l -- On order .....

Transversal T~ I In fact, C' ) Spec(A) ) sequence Conversely Spec(A(O1) to prove ×N)) transversal parametric to C', equations x 1 = x 1 x + a jl 1 2 x +. C': . ×. j 1 j2 = a l, jl × 1 2 +a I x +. =O'. , jI jI that for j . 4N. all i,j, C=C'. ) of the point O. , an irreducible i-th curve quadratic 1 transform of U . Sometimes, on the at O.. points 0 ,0 o I simplicity, , . . , 0 . , l . - For components and them, obtains from one C. , neighbourhood points points, may its i r r e d u c i b l e uniquely large in it.