Über einen Fundamentalsatz der synthetischen algebraischen Geometrie von W. Burau und H. Timmermann. (On a fundamental theorem in synthetical algebraic geometry of W. Burau and H. Timmermann)
The following theorem which is connected with the new book ''Algebraische Geometrie'' of \textit{W. Burau} and \textit{H. Timmermann} is proved: Theorem. Let X be a finite dimensional vector space over a commutative field K, let \(\hat X\) be the dual space of X, let \(s\in {\mathbb{N}}\) and \(X^ s:=\{x_ 1x_ 2...x_ s| x_ i\in X\setminus \{0\}\}\) the corresponding Segre manifold and let \(X^{(s)}:=\{xx...x| x\in X\setminus \{0\}\}\) be a Veronese manifold located on \(X^ s\). Then for any \(Y\in \hat X\) the sets \(YX^{s-1},XYX^{s-2},...,X^{s-1}Y\) are Segre submanifolds of \(X^ s\), and \(^ sY:=<YX^{s-1}\cup XYX^{s- 2}\cup...\cup X^{s-1}Y>\) is a hyperplane in the space \(<X^ s>\) generated by \(X^ s\). For \(A:=\cap \{^ sY| Y\in \hat X\}, B:=<X^{(s)}>\) we have \(<X^ s>=A\oplus B\) if and only if one of the following conditions is valid: (1) \(\dim X=1,\) (2) char\(K=0,\) (3) char\(K>s,\) (4) \(| K| \geq s,\dim X=2,\) \(0<p:=char K\leq (s+1)\) and \(s\equiv -1 mod p^ n\) where n is determined by \(p^ n\leq s+1< p^{n+1}\).
scientific article; zbMATH DE number 3986070