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SI,TI) = O we see L = L{SI,TI), i = 1, 2, 3. Then necessarily 0 = \dx{d,Pi{X,μ; SI, TI)) = θχΡ^{Χ, μ; SI, TI) · d,P^{X,μ; SI,TI), So Δ ( λ , μ ) or A{SI,TI) = 0. In fact Δ ( λ , μ ) = A{SI,TI) = 0, cf. Section 1. And conversely: If Δ ( λ , μ ) = 0 the tangent at χ{Χ,μ) touches S again at χ ( λ ' , μ ' ) corresponding to the other root (Χ',μ') of Pz(-,-;SI,TI). 5) it corresponds to a trisecant L* of S' on the tangent scroll of S'. Since S' generates P 3 , L* coincides with one tangent to S'.

Let : mon(2,m) —> K[X] be the map obtained by putting q*{t) = X* for all ¿ G mon(2,m) where X'= Π yGctib(2,m) ^f^'^ y(i),y(2)* From Theorems 6 and 8 we get a proof of the following Theorem 9 which, among other things, says that g|stab(2,m) is ÜT-basis of K[X]] we may call this the standard basis of K[X]\ from Theorems 4 and 8 we can deduce the following Theorem 10 which, among other things, says that this basis partitions well for various ideals in K[X]; Theorem 10 also says that q* provides bases modulo the said ideals.

Considermg Ap and Rp instead of A and R respectively, we may assume that A is a local ring. , yx^^)r-' and φ 2 Φ ^ x^T or φ ^ yx^T for some i. Without loss of generality we may assume φ ^ xl^'T or φ ^ yxlT, Suppose φ ^ xl'^T. We put U = xl^^T, S = R[l/U], Β = So and Ü = ΠΒ (D mJ5). Since S = B[U, l/U] and U is algebraically independent over B, S^s is Cohen-Macaulay if and only if so is B^. The Cohen-Macaulayness of B^ follows from [7, Satz 3] because the elements a : i , . . , Xd/xi, y/xl] is the coordinate ring of an afiine chart of Y, the Cohen-Macaulay scheme given in [7, Satz 3](cf.

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