Babylonian tower theorems for coverings
Let \(Y_ n\), \(n\geq n_ 0\), be a sequence of n-dimensional algebraic varieties over the algebraically closed field k, which are weighted complete intersections in the weighted projective space \({\mathbb{P}}(e_ 0,...,e_{n+r})\). Assume that \(Y_ n\) is the intersection of \(Y_{n+1}\) with \({\mathbb{P}}(e_ 0,...,e_{n+r})\) where \({\mathbb{P}}(e_ 0,...,e_{n+r})\) is considered as a subspace of \({\mathbb{P}}(e_ 0,...,e_{n+r+1})\) in the obvious way. Let \(X_ n\to Y_ n\) be a sequence of locally complete intersection morphisms such that \(X_ n=X_{n+1}\times_{Y_{n+1}}Y_ n\) for \(every\quad n.\) Then, generalizing a result of \textit{E. Sato}, the following is shown in the paper: If \({\mathcal O}_{Y_ n}(1)\) is invertible and \(X_ n\) is reduced and connected then \(X_ n\) is a weighted complete intersection in some projective space \({\mathbb{P}}(e_ 0,...,e_{n+r},b_ 1,...,b_ s)\) for every n. In particular, this applies to towers of coverings \(X_ n\) of the usual projective space \(Y_ n={\mathbb{P}}^ n\). For the proof the methods of a previous paper by the author [cf. Math. Ann. 271, 153-160 (1985; Zbl 0541.14011)], play an important role. As a byproduct there is also shown a generalization of the theorem of A. N. Tyurin and E. Sato on infinitely extendable vector bundles.
scientific article; zbMATH DE number 4142213