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On the dimension of the adjoint linear system for threefolds. (English) Zbl 0857.14001
Let \(L^\wedge\) be a very ample line bundle on a smooth complex projective 3-fold \(X^\wedge\). By previous work of the second author it is known that there exists an integer \(t>0\) such that \[ h^0(t(K_{X^\wedge} + L^\wedge)) \geq 1, \] except for a short list of pairs \((X^\wedge, L^\wedge)\) whose structure is well understood; moreover there exists a morphism \(r:X^\wedge\to X\) expressing \(X^\wedge\) as a smooth 3-fold \(X\) blown-up at a finite set and an ample line bundle \(L\) on \(X\) such that \(K_{X^\wedge} + 2L^\wedge = r^*(K_X+2L)\); \(K_X+2L\) is ample and \(K_X+L\) is nef and big. Pairs \((X^\wedge,L^\wedge)\) satisfying these conditions are said to be of log-general type. There is a hope, supported by several results, that \(K_X+L\) is in fact spanned except for a few cases.
In connection with this in the paper under review the authors study the dimension of the linear system \(|K_X + L|\) (or equivalently of \(|K_{X^\wedge} + L^\wedge|)\) proving the following. If \((X^\wedge, L^\wedge)\) is of log-general type then \(h^0(K_{X^\wedge} + L^\wedge)\geq 2\); moreover, if \(X^\wedge\) has Kodaira dimension \(\kappa(X^\wedge)\geq 0\), then \(h^0(K_{X^\wedge} + L^\wedge)\geq 5\) with equality if and only if \((X^\wedge, L^\wedge)\) is a smooth quintic hypersurface of \(\mathbb{P}^4\). Of course these results have consequences for higher dimensional smooth projective \(n\)-folds polarized by a very ample line bundle \({\mathcal L}\), provided that \(|{\mathcal L}|\) contains \(n-3\) elements meeting transversally along a smooth 3-fold.
Reviewer: A.Lanteri (Milano)

MSC:
14C20 Divisors, linear systems, invertible sheaves
14J30 \(3\)-folds
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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