By M. F. Atiyah, I. G. MacDonald

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1jJ(s). If Q is a moving Proof 1) Assume 1jJ is constant and take a basis (vo, ... , Vk) of V vector on U C S, then there are hoI omorphic functions Ai on U such that k k Q=LAiVi' hence Q' = L A; Vi i=O E V, i=O and this implies 1jJ(l) C 1jJ. Conversely, if 1jJ(I) = 1jJ then for every local basis (Qo, ... , Qk) on U we have holomorphic functions A{ on U such that k Qj = L A{ Qi for j = 0, ... ,k. i=O We consider the holomorphic maps " . U ~ ~ <>: N+1, IAk+1

Proposition. Given a compact Riemann suiface S and a holomorphic curve tfr : S ~ G(k, N) of drill d, there exist holomorphic curves tfr(1) : S ~ G(k + d, N) , tfr(l): S ~ G(k - d, N) such that tfr(l)(s) = tfr+(s) and tfr(l)(s) = tfr-(s) for almost every s E tions are dual in the following sense: s. These construc- Proof We take a point a E S and a neighborhood U with a distinguished local basis QO, ... ,Qk as in the Main Lemma. On U \ {a} we have the equalities tfr+ = span {Qo, ... ,Qk. Q~, ...

4, the map The holomorphic curve cp n l{f can be derived from a holomorphic sum by duality, since The sum cp + l{f is called direct if m = k + 1 + 2, which is equivalent to n = 0; we write 1 Grassmannians 44 Using this concept of direct sums, we can show the existence of sufficiently many directrices. Lemma. Given a compact Riemann surface S and two curves 1/1: S --+ f{J: S --+ G(k, N), with f{J C G(l, N) 1/1, there exist directrices f{Jl, ... ·. ,vk, WI, ... , W/-k, ul,· .. - such that the first k + 1 vectors are a base of f{J(0) and the first 1 + 1 vectors are a base of 1/1 (0).

### Introduction to Commutative Algebra by M. F. Atiyah, I. G. MacDonald

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