Hypersurfaces of inifnite dimensional Banach spaces, Bertini theorems and embeddings of projective spaces
This short paper in infinite dimensional algebraic geometry studies Bertini type problems and embeddings of projective spaces via clever methods based on hypersurfaces, lines and curves. The following three theorems are proved. Theorem~1. Let \(V\) be an infinite dimensional Banach space, \(X\subset P(V)\) a finite codimensional closed analytic subset of the projectivization \(P(V)\) of \(V\). If \(X\) itself is not a linear subspace of \(P(V)\), but \(X\) contains a finite codimensional linear subspace \(M\) of \(P(V)\), then \(X\) is singular, and its singular locus \(\text{Sing}(X)\) contains a closed finite codimensional analytic subset \(T\) of \(M\). Theorem~2. Let \(V,P(V)\) be as in theorem~1, \(A\subset V\) a finite dimensional linear subspace, and \(Y\) a closed analytic hypersurface of \(P(V)\). Then there is a linear subspace \(B\subset V\) such that \(A\subset B\), \(\dim(B)=\dim(A)+1\), and \(\text{Sing}(Y)\cap A =\text{Sing}(Y\cap B)\cap A\). Theorem~3. Let \(V\) and \(E\) be infinite dimensional complex Banach spaces, \(j:P(V)\to P(E)\) a closed embedding with \(j(P(V))\) a finite codimensional analytic subset of \(P(E)\). Then \(j\) is a linear isomorphism onto a finite codimensional closed linear subspace of \(P(E)\). The paper is well written and easy to follow.
scientific article; zbMATH DE number 2065403