°DI CLAIM: The canonical morphism, if and only if either 1:1>* is an isomorphism D is non -singular at P or D does not pass through P.

1. Choose H as before. H nH 1 nC ::~. Let f Thus g is surjective and d 2 :: O. If n ~ X+ 1 then, d n >d n + 1 or d n :: O. Now choose another hyperplane HI such that = 0 and g = 0 be their respective equations. >O 1 1 1o o o The bottom sequence is exact because, f and g hAve no common zeros on C. O ~ ~ p p M(n+1) /' o is exact, ~0 From this we get a complex, HI (J(n-l» ---'> HI (<)p(n-l >EBJ(n)E9J(n» ---+Hl PIP(n) EB qp(n)~J(n+l)). Denote the homology at the middle by Wn + 1 • 2, CLAIM. Wn +1 = Coker (Ho(J(n+l)~~(n)~ HO (Op(n+I)), exact sequences as follows for n ~ We have X + I, o i O--~ Wn +1--+ H1 (M(n+l)) --+H1 (J(n+l)) --+ 0 i H 1 (J(n)'+ J(n» T.

### Lectures on Equations Defining Space Curves by L. Szpiro (auth.)

by Jeff

4.5