Extension of modifications of ample divisors on fourfolds: II
[For part I see ibid. 36, 106-120 (1984; Zbl 0513.14004).] Let X be a connected 4-dimensional projective manifold of nonnegative Kodaira dimension. Let A be a smooth ample divisor on X with A the blow up of a smooth projective manifold A' along \(P^ 1\), the projective line. Then \((i)\quad if\) \(Y\not\simeq P^ 1\times P^ 1\) (Y is the exceptional divisor on A) there exists a 4-dimensional manifold X' such that A' lies in X' as a divisor and such that X is the blow up of X' along \(P^ 1\); \((ii)\quad if\) \(Y\simeq P^ 1\times P^ 1\) then (i) is still true unless \(N_{A/X,Y}={\mathcal O}(a,1)\) with \(a\geq 2\). In the latter case there exists a 4-dimensional Cohen-Macaulay variety and a morphism \(\phi:\quad X\to X'\) such that \(\phi\) maps X-D biholomorphically onto \(X'-P^ 1\quad ((D,\phi)\) is a \(P^ 2\)-bundle over \(P^ 1).\) If X is a local complete intersection with isolated singularities then (i) and (ii) are still true but in the case (i) X' is a local complete intersection.
4-dimensional projective manifold of nonnegative Kodaira
4-dimensional projective manifold of nonnegative Kodaira dimension
scientific article; zbMATH DE number 3961832